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Metody opisu dyfuzji wielu składników, unifikacja metody dyfuzji wzajemnej i termodynamiki procesów nieodwracalnych Marek Danielewski. Interdisciplinary Centre for Materials Modeling AGH Univ. o f Sci. & Technolog y , Cracow, Poland Będlewo, Czerwiec 2013. φ. φ. Quantum mechanics:. - PowerPoint PPT Presentation
Citation preview
Metody opisu dyfuzji wielu składnikoacutew
unifikacja metody dyfuzji wzajemnej i
termodynamiki procesoacutew nieodwracalnych
Marek Danielewski
Interdisciplinary Centre for Materials Modeling
AGH Univ of Sci amp Technology Cracow Poland
Będlewo Czerwiec 2013
Diffusion equation (Fourier)
2
2
Fundamental or only numerology
t x
Diffusion equationsHeat
T t
Θ 2 T
x 2
Θ = α m2s-1
Diffusion of mass (1855)
t
Θ 2 x 2
Θ = D m2s-1
Diffusion equations
Hydrodynamics (noncompressible fluid)
υ t
Θ 2υ
x 2
Θ = ν m2s-1
Diffusion equations
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Diffusion equation (Fourier)
2
2
Fundamental or only numerology
t x
Diffusion equationsHeat
T t
Θ 2 T
x 2
Θ = α m2s-1
Diffusion of mass (1855)
t
Θ 2 x 2
Θ = D m2s-1
Diffusion equations
Hydrodynamics (noncompressible fluid)
υ t
Θ 2υ
x 2
Θ = ν m2s-1
Diffusion equations
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Diffusion equationsHeat
T t
Θ 2 T
x 2
Θ = α m2s-1
Diffusion of mass (1855)
t
Θ 2 x 2
Θ = D m2s-1
Diffusion equations
Hydrodynamics (noncompressible fluid)
υ t
Θ 2υ
x 2
Θ = ν m2s-1
Diffusion equations
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Diffusion of mass (1855)
t
Θ 2 x 2
Θ = D m2s-1
Diffusion equations
Hydrodynamics (noncompressible fluid)
υ t
Θ 2υ
x 2
Θ = ν m2s-1
Diffusion equations
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Hydrodynamics (noncompressible fluid)
υ t
Θ 2υ
x 2
Θ = ν m2s-1
Diffusion equations
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Quantum mechanics
2
2t x φφ
2
i
m
2 2
2
i
m
i
2 1i
i
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Quantum mechanics
2 2
22i
t m x
free particlehellip
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Question
Why
2
i
m
Answerhellip P-K-C hypothesis
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Economyhellip
hellip diffusion equation
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997
for a new method to determinehellip the value of derivatives
Robert C MertonHarvard University
Myron S ScholesLong Term Capital Management Greenwich CT USA
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Economy amp diffusionhellipButhellip money is not conserved
Merton amp Sholes Nobel price bdquohelped inrdquohellip
hellipgrand failure in 2008
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
2
2t x
lt 10-35 - Planck scale- 10-35
nucleus - 10-16
atoms - 10-10
biology - 10-4
mechanics - 1 Earth - 107
cosmology ndash 1027
gt 1030
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Fundamental
2
2 t x
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Challengeseverywherehellip
Mechano-chemistry
Darken amp stressUniquenesshellip
Electro-chemistry
Nernst-Planck- Poisson + drift
AppliedReactive inter-diffusionhellipReal geometryhellip
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
grad
div dii i
di i jj
ch elj i ij
cc
t
B F
F
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
div i iiz cE F
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Nernst-Planck-Poisson Problem
1( ) 2( ~ 1900)
3(~ 1960) 4
i Nernst Planck
Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840
Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941
MaxMax PLANCK PLANCK 1858-19471858-1947
Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip
W Kucza (2009) convergehellip
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Nernst-PlanckNernst-Planck
di i jj
d di i iJ c
B F
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip
d drifti i iJ J c
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)
( ) iic c t x const
Showhellip
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
( ) iic c t x const
No stresshellip
Ωi = Ω = const
R1hellip
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
t
d D DD
dt Dt Dt
Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative
gradD
Dt t
or
material velocity = or
m
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
gradM
MD
Dt t
local centre of composition
M iii
c
c
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
gradm
mD
Dt t
local centre of mass
m iii
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
gradD
Dt t
local volume velocity
1
r
i ii
i
c
c
None of them
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
( ) iic c t x const
If not
Then
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
EOS Vegard law
( ) 0dii
J t x ( ) 0ii
J t x ( ) 0i ii
z FJ t x ( ) 0d drift
i iiJ t x c
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
We need different approachhellip
Darken
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Bi-velocityhellip
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Lattice sites not conserved
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip
to the targetto the target
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
~250 ag onl o y mEuler
Oumlttinger (2005) bdquosomething is missingrdquo
19th century Cauchy Navier Lameacutehellip
Stephenson (1988) drift amp m
up to 2007 only m
Brenner (2006) Fluid Mechanics Revisitedhellip
Cracow (1994) vd amp drift
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)
1 Complemented volume fixed RF
2 Was polite to not notice
conflict between RFrsquos
hellip in our papers
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
150 years of diffusion equation
Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)
Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)
Lattice sites are not conserved (1948 Kirkendall amp Darken)
Diffusion velocityhellip (~1900 Nernst amp Planck)
Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)
Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
150 years of diffusion
Number of laws decreaseshellip
Complexity increaseshellip
Do we bdquostay withrdquo m ρυ q U only
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
B100
A100
A(x)
o
ΛΛ
t= 0+
xxmλ1(0)
Dynamics amp diffusion
1 mx f t x
Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
B100
A100
t=0+
xxmλ1(0)
a)
Λ Λ
A(x)o
A(x)tgt0b)
xλ1(tgt0) xm
c) tA = const
xλ1(t) xm
λ1(0)
Dynamics amp diffusion
Yes
1
mx const
tf t x
1 mx f t x
Central problem
Eg diffusion anddeformation stressreactions
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Hopelesshellip fundamentals only
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
1 1 mr rf k x k x k f x x
1 11
r m
i rii
f k x k xx mk f x x
k x
Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the
variables x1hellip xr if
several identities follow eg
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
1 i
r iip T N N
NN
1 r ii ip T N N N
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
whole mixture 1
-th component
V
Vi i i
c
i c
Volume densities
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
From Euler theorem
dd dd d
d d di iit t
tc x c x
t t t
1
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The molar volume is the nonconserved property Buthellip is transported by components velocity field
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Fundamentals II
The Liouville transport theorem
dd div d
di
i i
t t
i
ff fx x
t t
fi is a sufficiently smooth function (eg have first
derivative C1) and υi is defined on fi
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Liouville
Conservation of component (fi = ci)
div
dd d 0
d i
t
i i
t
icc x
tc x
t
div 0ii i
cc
t
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The Liouville theorem amp the Volume Continuity
dd div d
di ii
i i i i ii it t
cc x c x
t t
fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)
ddiv d
d i i iit
c xt
1 1
div 0i i ii
constc
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
div 0i i iic
The volume density conservation law orhellipequation of volume continuity at constantvolume
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Overall drift velocity
drift D T tr drift d
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Finally due to Liouvillethe bi-velocty method
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
dVolume continuity div
d i i iic
t
Conservation of mass div 0 for 1 2ii i i r
t
Momentum conservation DivD
rD
g adi
extiii
Vt
D 1 grad d
D 3ivEnergy conservation
i
mi
i
q
i i ii i
up
tJ
I
drift di i
Entropy production divDD
i
ii
q
isp A
st
JT
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Entropy production divDD
i
ii
q
isp A
st
JT
1 1grad grad grad grad 0
ms d q di i i
i i i i i ii i i
pA u J
T T T T
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
11 12
21 22
grad ln
gradM
L Lq T
L LJ
L-matrix is both as required by LITsymmetric and non - negative
Brenner 2009 Danielewski amp Wierzba 2010
bull Entropy production term is always positive
bull Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Conservation laws in material reference frame
Bi-velocity vs LIT
Diffusion stress reactionsamp more
Planck-Kleinert Crystal
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Reactions amp Interdiffusion Multiples
T= 150 180 200oC
t = 3h 30h 100h 7days 14dayshellip
Experiment
Cu-Sn-Ag Cu-Ag-Sn-Ni
Fabrication and vacuum annealing
020cm 300cm
350deg
Cu Ag Ni Sn
M Pawełkiewicz EMPA amp AGH
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
International PhD School Switzerland ndash Poland
Fabrication Sectioning
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
International PhD School Switzerland ndash Poland
after heat treatment t=4h and T=180 C
Sn
Cu
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Interdiffusion amp stress
mechano-chemistry
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
InterdiffusionInterdiffusion
C o5 2 w t N i t= 0F e
5 1 w t N i
D iffu s io n co u p le
Ni-Cu Ni-Fe
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Model of InterdiffusionModel of Interdiffusion
tgt0
Diffusion couple
c(tx)
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
DDiv grad
De p extV
t
B Wierzba 2008
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Future
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Bi-velocity methodhellip at the Planck scale
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Oliver Heaviside (1850-1925)
bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated
Mathematics is an experimental science and definitions do not come first but later on
I do not refuse my dinner simply because I do not understand the process of digestion
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Planck-Kleinert Crystal
M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Soccer Balls Diffract
httpwwwquantumunivieacatresearchc60
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Professor Anton Zeilinger
Experiment amp theory for C60 and C70
C60F48 world record (108 atoms) in matter interferometry
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512
Already onhellip
bdquoPlanck-Kleinert Crystalrdquo
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Physics Today [1] rarr ldquoThe persistence of etherrdquo
Statistical mechanics [2] rarr dimensions become large
quantum properties emerge
Quantum space [3] rarr analogous to crystal
Kleinert [4] rarr Einstein gravity from a defect model
[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)
Vacuumhellip No [5]rarr ldquoThere is no information without
representationrdquo
[5] W Żurek Nature 453 (2008) 23
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Volume continuity div 0c
Conservation of mass div 0c
ct
DMomentum conservation Div grad
D m
mextV
t
DI law Grad div
D qJt
ddiv
d i i iic
t
div 0 for 1ii i
cc i r
t
DDiv grad
Di
extiii
Vt
D 1 grad div
D 3i
ii i i i qi i
p Jt
I
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)
- Frenkel disorder
- defects form solid solution - defects diffuse
- bdquoclassicalrdquo conservation laws
- volume continuity amp material reference frame
- double valued deformation field
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
divt
DDiv
D m
m
t
DDiv
D
e
t
Volume continuity
Mass conservation
Navier-Lame+ diffusion
Energy conservation
Div 2 graddiv divgrad u uσ σ
P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip
div 0c
m d
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Included the entropy production as a result of defect formation and diffusionhellip
L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
+ stationary travelingamp
their combinations
Processes
1 Transverse wave
2 Longitudinal wave
3 lattice deformation (Kleinert 2003)
4 Pi diffusion (mass)
5 Heat transferhellip
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference
Lattice parameter Planck length lP 161624(12)10-35 m NIST
Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson
Mass of particle Planck mass mP 217645(16)10-8 kg NIST
Frequency of the internal process
Inverse of the Planck time
fP = 1tP 185486(98)1043 s-1 NIST
Lameacute constant Energy density 185323719410114 kgm-1s-2 This work
Number of particles in unit cell 4 This work
National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)
The physical constants (ideal regular fcc lattice)
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Rem
Imm
Re Imm m mi
The energy of volume deformation field
The energy of the torsion field
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
divt
DDiv
D
e
t
Mass conservation
Energyconservation
Volume continuity div 0d
Gravity
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
[defects] asymp const at T= const
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
in Planck-Kleinert crystal
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
div m dP P e
already Newtonhellip
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Redivgrad 4mMG
2where P PG l c m
Simeacuteon-Denis Poisson
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
116674189 10G NIST data G = 66742(10) bull 10-11
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
4 The ldquodark energyrdquo rarr energy of the DIPPrsquos
1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos
2 DIPPrsquos create the gravitational interaction between matter
3 The ldquodark matterrdquo rarr DIPPrsquos
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Remark 1 Planck length = Schwarzschild radius
2 P WIMP gravitonm m m m Higg s boson
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Electromagnetism
Zero diffusion
So far
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
042223
Dgraddiv divgrad
DL L
P PP Pm f m ft
σ σNavier-Lame
amp no diffusion
Only transverse wave
1) 0d
2) ρ = ρ0 const
3) div 0
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
3 3
2
2 = 8 grad 8 grad divdiv
P P
P P
L Ll lm mt
xx + x
2
2
2 2 = rot rotLc ct
x
x + x
Transverse waves in P-KC
Equivalent form
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
2
3 1
2
2 = rotr
8
ot1
km299 792 s
P PLc l
t
m
c
xx
Equation of the transverse wave
299 792 5 (NIST)
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
= rotrotm Lt
x
B
t
0
B
t
rotE
mB
22
rot rotLP
P
ff
x
20
1 L
Pf
2rotPfE x
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Full set Maxwell eqs in vacuum
analogous simple transformations
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
divt
DDiv
D
e
t
Mass conservation
EnergyConservation
andhellip
Quantum mechanics
Re Imm m mi
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Imm
Im Imgrad grad d m mM M M MJ B eB
The energy flux
1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of
particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Im
div
gradd mM M
ee
t
J eB
Imdiv grad mM
eB e
t
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Im
0Re Im 0
div grad
exp 22
mM
m m
M P P
eB e
t
e i
B m M B
Re Im Re ImIm2 2
exp 2 div exp 2 gradm m m m
mP Pi im B
t c M c
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Re Im Re ImIm2 2
2Re Im
exp 2 div exp 2 grad
exp
m m m mmP P
m m
i im B
t c M c
i c
2
2Imdiv grad mP Pm B
t M
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
2
div grad 2P Pm B c
Mi E
t
22 P Ph m B c
div4
grad i Et
h
M
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The physical constants at the Planck scale and
four time scalesPhysical Quantity Value in SI units SI unit Reference
Volume of PC cell 4222 002 82810-105 m3 This work ampNIST
Planck density 2062 008 6621097 kg m-3 This work amp NIST
Young modulus 4633 092 98610114 kg m-1s-2 This work
Planck mass mobility 5391 213 98210-44 s This work
Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work
Planck constant6626 069 31110-34
6626 069 3(11)10-34 kg m2 s-1 This workNIST
Gravitational constant66742(10)10-11
667417610-11 m3 kg-1 s-2 NISTThis work
Speed of longitudinal wave 519 255 240 m s-1 This work
Speed of transverse wave299 792 153299 792 458
m s-1 This workNIST
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Physical reality at the Planck scale
bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip
consider
bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales
bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity
bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows
3L c
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
bull Transverse wave equiv electromagnetic wave
bull DIPPrsquos rarr Dark Matter rarr Dark Energy
bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Conclusions
Fluxes rarr Nernst-Planck formulae
Collective behavior rarr standing wave (rdquoparticlerdquo)
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Conclusions
Multi-phase and
multi-component
Today in R1
Tomorrowhellip R3
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Future
bull new experimental methods
bullnew processes to predict
bull all methods developed in math and physics will be usefull
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Forthcoming
elec
tro-mechano-chemistry
d
i i
eli
chi
miB
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Planck-Kleinert Crystal straightforward
vs
Remark
Complexity of diffusion processes in multicomponenthellip systems
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
E N D
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
Fundamentals I
Euler - the volume amp molar volume
1 1 r rkn kn T p k n n T p
hellip homogeneous of the 1st degree in the variables n1hellip nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr
The volume and molar volume
Ni = cic is molar ratio
1 1 r rkn kn T p k n n T p
1 1 r rN N T p k n n T p
hellip homogeneous of the 1st degree in the variables N1hellip Nr