Temperature of moving bodies – thermodynamic, hydrodynamic and kinetic
aspects
Peter VánKFKI, RMKI, Dep. Theoretical Physics
– Temperature of moving bodies – the story– Relativistic equilibrium – kinetic theory– Stability and causality – hydrodynamics– Temperature of moving bodies – the conclusion– Outlook
with Tamás Biró, Etele Molnár
Planck and Einstein
body
vobserver
K0
K
pdVTdSdE
Relativistic thermodynamics?
About the temperature of moving bodies (part 1)
• Planck-Einstein (1907): cooler
• Ott (1963) [Blanusa (1947)] : hotter
• Landsberg (1966-67): equal
• Costa-Matsas-Landsberg (1995): direction dependent (Doppler)
0TT
)cos1(0
v
TT
v
body
observer
K0
K
0T
T
0TT
21
1
v
0
0
0
0
0
/
vdEdG
dEdE
dSdS
dVdV
pp
vdGpdVTdSdE
translational work – heat = momentum
00000
000
0
020
000
dVpdSTdE
dVpTdS
dE
dEvdV
pTdSdE
vdGpdVTdSdE
vobserver
K0
K
reciprocal temperature - vector?
0TT
21
1
v
Rest frame arguments: Ott (1963)
v
body
reservoir
KK0
dQ
Planck-Einstein
v
body
reservoir
K
K0
dQ
Ott
vdGpdVTdSdE
0
0
0
02
/
dEdE
dSdS
dVdV
pp
00000
002
00 /
dVpdSTdE
dVpTdSdE
pdVTdSdE
v
body
observer
KK0
0TT
No translational work
Blanusa (1947)Einstein (1952) (letter to Laue)
temperature – vector?
Outcome
T
ku
ekxf
),(0 0T
T Einstein-Planck (Ott?)
Relativistic statistical physics and kinetic theory:
Jüttner distribution (1911):
Historical discussion (~1963-70, Moller, von Treder, Israel,
ter Haar, Callen, …, renewed Dunkel-Talkner-Hänggi 2007): new arguments/ no (re)solution.
→ Doppler transformation e.g. solar system, microwave background
→ Velocity is thermodynamic variable?
Landsberg
van Kampen
Questions• What is moving (flowing)?
– barion, electric, etc. charge (Eckart)
– energy (Landau-Lifshitz)
• What is a thermodynamic body?– volume– expansion (Hubble)
• What is the covariant form of an e.o.s.?– S(E,V,N,…)
• Interaction: how is the temperature transforming
→ kinetic theory and/or hydrodynamics
Boltzmann equation)( fCfk
Kinetic theory → thermodynamics
'' 11 ffff
kxxekxf )()(
0 ),( (local) equilibrium distribution
)1(ln
0
3
ffkk
kdS
Thermodynamic equilibrium = no dissipation:
2mkk
Boltzmann gas
01ln4
1|0
3
0
3
0
3
,,,0
3
klijji
ji
lk
ji
lk
l
l
k
k
j
j
lkji i
i Wffff
ff
ff
ff
k
kd
k
kd
k
kd
k
kd
T
ku
ekxf
),(0T
u
T
,
Thermodynamic relations - normalization
Jüttner distribution?
kxxekxf )()(
0 ),( 00 fkN
00 fkkT
kkfkffkN 0000
000 TNN
000 )1(: TNS
000 TNS
Legendre transformation
0000
TNS
0
TNS
covariant Gibbs relation(Israel,1963)
Lagrange multipliers – non-equilibrium
qeuE
PuqEuT
jnuN
JsuS
Rest frame quantities:
.0,0
;0,0
;,1
uPPuqu
juJu
uuuu
0)(
00
Puqj
uEnsEns
TNS
Remark:
0,0,0 wT
0)(
PuqjuEnsEns
0 Ens
T
wu
T
g
E
s
Tn
s
Ens
),(
0wu
0)(
PuqjuEns
nTEnTsp ideal gas
rest frame/uniform intensives
duqewdqwdeqeudwudEdnTds )()()(
Velocity dependence?
deviation from Jüttner
A)
B)
Energy-momentum density:
T
ku
T
kwuk eeekxf ˆ
ˆˆ)(
0 ),(
21
ˆw
wuu
,1
ˆ,1
ˆ22 w
T
w
TT
nwnuunfkN ˆˆ00
TeT
mKTmwnn
ˆˆ41ˆ 2
22
00ˆˆˆˆˆ fkkpuueT
T
mK
T
mK
nmTne
ˆ
ˆˆˆˆ3ˆ
2
1
pe
qqpquuqueuT
0
wpeqw
wpeepp )(,
1
ˆˆ,ˆ
2
2
Heat flux:0
w
n
peqI
Summary of kinetic equilibrium:
- Gibbs relation (of Israel):
- Equilibrium spacelike parts:
nwnuN 0
pe
wwpwuuwpeueuT
)(0
dEwudEgdnTds )(
• What is dissipative? – dissipative and non-dissipative parts
• Free choice of flow frames? – QGP - effective hydrodynamics.
• Kinetic theory → hydrodynamics – local equilibrium in the moment series expansion
→ talk of Etele Molnár
• What is the role and manifestation of local thermodynamic equilibrium?
– generic stability and causality
Questions
Nonrelativistic Relativistic
Local equilibrium Fourier+Navier-Stokes Eckart (1940),(1st order) Tsumura-Kunihiro
Beyond local equilibrium Cattaneo-Vernotte, Israel-Stewart (1969-72),(2nd order) gen. Navier-Stokes Pavón, Müller-Ruggieri-Liu,
Geroch, Öttinger, Carter, conformal, Rishke-Betz,etc…
Eckart:
Extended (Israel–Stewart – Pavón–Jou–Casas-Vázquez):
T
jqunesNTS
),(),(
qqjqT
uT
qqTT
nesNTS
10
2120
1
222),(),(
(+ order estimates)
Thermodynamics → hydrodynamics (which one?)
Israel–Stewart - conditional suppression (Hiscock and Lindblom, 1985):
qqjqT
uT
qqTT
nesNTS
10
2120
1
222),(),(
,0)(
11
enn
s
np
nep
pe
T
p
e
pe,0...
)/()/(
12
nTp
ns
p
ns
e
pe
,02
,0,02
21
172805
,0)/(
1
3
22
2
21
0
20
16
n
ns
T
Tn
,0
2
222121
1124
pe
,03
211)(
6
2
203
K
e
ppe
n
s .0
3
21
/2
1
0
0
nsn
T
T
nK
Remarks on causality and stability:
Symmetric hyperbolic equations ~ causality
– The extended theories are not proved to be symmetric hyperbolic (exception: Müller-Ruggeri-Liu).
– In Israel-Stewart theory the symmetric hyperbolicity conditions of the perturbation equations follow from the stability conditions.
– Generic stable parabolic theories can be extended later.
– Stability of the homogeneous equilibrium (generic stability) is related to thermodynamics.
Thermodynamics → generic stability → causality
02)(
,0
~~4
2
~~2~~~~
Tc
v
c
vTv
TT
tttxxxxt
xxt
aa
a jTT
qJ
0),(
JusnesS
junnN
PuququeeTu
0
Special relativistic fluids (Eckart):
0)()(1
2
uTT
T
qupP
TTj
Eckart term
., jnuNPuququeuT
011
TqvpP
T ii
jiijij
qa – momentum density or energy flux?
ijj
i
Pq
qeT
Bouras, I. et. al., PRC 2010 under publication (arxiv:1006.0387v2)
Heat flow problem – kinetic theory versus Israel-Stewart hydro in Riemann shocks:
0),( aa
aa
aa JusnsS
Improved Eckart theory:
junnN
PuququeeTu
0
EEe 22 qInternal energy:
01
2
e
qTuTT
T
qupP
TTj b
Eckart term
aa
a jTT
qJ
Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)
duqewdqwdedEdnTds )(
Dissipative hydrodynamics
< > symmetric traceless spacelike part
.2
,
,
,
,0)()(
,0)(
,0
u
upP
T
e
qTuTTq
quququpeT
uuqqupeeTu
junnN
vv
vvv
vvv
vv
vvv
v
linear stability of homogeneous equilibriumConditions: thermodynamic stability, nothing more.
Ván P.: J. Stat. Mech. (2009) P02054
Israel-Stewart like relaxational (quasi-causal) extensionBiró T.S. et. al.: PRC (2008) 78, 014909
integrating multiplier
Hydrodynamics → thermodynamics
}3,2,1,0{,
,,,1
GuEE
dVqGeVedVEdVuV
u
QdAquVupGuE
HH H
H
H(2)
H(1)
Volume integrals: work, heat, internal energy
Change of heat and entropy:
pdVAA
uAdG
AA
AdE
AA
uAdS
VESSGuEE
ASVupEQ
),(,
pdVdEgTdS
temperature1
ug
integrating multiplier
• there are four different velocities• only one of them can be eliminated • the motion of the body and the energy-momentum currents are slower than
light
pdVdEgTdS
ugwug 1
Interaction
v2
observer
w2
v1
w1
w spacelike, but |w|<1 -- velocity of the heat current
About the temperature of moving bodies (part 2)
v2
w2
v1
w1
2
222
1
111
2
222
1
111
T
)wv(
T
)wv(
T
)wv1(
T
)wv1(
2
2
2
1
2
1
22
22
11
11
T
w1
T
w1
wv1
wv
wv1
wv
2
2
1
1
T
g
T
gdEgTdS
1+1 dimension:
),(),,( wvwwvu
2
2
2
1
1
1
vw
v
T
T
Four velocities: v1, v2, w1, w2
Transformation of temperatures
2
22
1
21
22
22
11
1111
,11 T
w
T
w
wv
wv
wv
wv
v
w2w1Relative velocity (Lorentz transformation) 21
12
1 vv
vvv
general Doppler-like form!
2
21 1 vw
wvw
0
2
0 1
1
vw
v
T
T
Special:
w0 = 0 T = T0 / γ Planck-Einstein
w = 0 T = γ T0 Ott
w0 = 1, v > 0 T = T0 • red Doppler
w0 = 1, v < 0 T = T0 • blue Doppler
w0 + w = 0 T= T0 Landsberg
v
w0w
K K0thermometer
T T0
Biró T.S. and Ván P.: EPL, 89 (2010) 30001
Summary
Generalized Gibbs relation:
– consistent kinetic equilibrium– improves hydrodynamics– explains temperature of moving bodies
KEY: no freedom in flow frames (Eckart or Landau-Lifshitz)!?evolving frame?is dissipation frame independent?QGP - effective hydro
Outlook:Outlook:
Dissipation beyond a single viscosity?Dissipation beyond a single viscosity?Causal and generic stable hydro from improved moment series expansion.
dEgdnTds
u1
u 2
w 1
w 2
g 1
g2
Blue shifted doppler
u 1
u 2
w 1 w 2
g1
g2
Planck-Einstein
u1
u 2
w 1
w 2
g1 g2
Landsberg
u 2
w 1
w 2
g1 u1g2
Ott-Blanusa
u1
u 2
w 1
w 2
g 1
g2
Red shifted doppler
.2
,j
iijk
kij
ii
vv
Tq
Isotropic linear constitutive relations,<> is symmetric, traceless part
Equilibrium:
.),(.,),(.,),( consttxvconsttxconsttxn ii
ii
Linearization, …, Routh-Hurwitz criteria:
00)(
,0
,0,0,0
TT
TTpnpp
T
nnn
,0
,0
,0
iijj
jj
ii
jiiji
ii
i
ii
Pvkk
vPqv
vnn
Hydrodynamic stability )( 22 sDetT
Thermodynamic stability(concave entropy)
Fourier-Navier-Stokes
0)(11
jiijij
ii vnTsP
TTq
p
Remarks on stability and Second Law:
Non-equilibrium thermodynamics:
basic variables Second Lawevolution equations (basic balances)
Stability of homogeneous equilibrium
Entropy ~ Lyapunov function
Homogeneous systems (equilibrium thermodynamics):dynamic reinterpretation – ordinary differential equations
clear, mathematically strictSee e.g. Matolcsi, T.: Ordinary thermodynamics, Academic Publishers, 2005
partial differential equations – Lyapunov theorem is more technical
Continuum systems (irreversible thermodynamics):
Linear stability (of homogeneous equilibrium)
Thermodynamics
Hydrodynamics Kinetictheory
homogeneity equilibrium
moment series
general balances
concepts
concepts homogeneity
Summary
– S = S(E,V,N)– Work with momentum exchange– Relative velocity v is zero– Cooler, hotter, equal or Doppler?
Ván: J. Stat. Mech. P02054, 2009.Bíró-Molnár-Ván: PRC 78, 014909, 2008.
Bíró-Ván: EPL, 89, 30001, 2010, (arXiv:0905.1650)Wolfram Demonstration Project, Transformation of …
Internal energy:
E Ea
– S = S(Ea, V, N)– energy-momentum exchange– T and v do not equilibrate– γwT and w v are equilibrating
– T: Doppler of w with the speed v
Heavy ion physics: dissipative relativistic fluids