Weakly nonlocal continuum physics – the role of the Second Law

Peter Ván HAS, RIPNP, Department of Theoretical Physics

– Introduction • Second Law

• Weak nonlocality

– Liu procedure

– Classical irreversible thermodynamics

– Ginzburg-Landau equation

– Discussion

general framework of anyThermodynamics (?) macroscopic

continuum theories

Thermodynamics science of macroscopic energy changes

Thermodynamics

science of temperature

Nonequilibrium thermodynamics

reversibility – special limit

General framework: – Second Law – fundamental balances– objectivity - frame indifference

Thermo-Dynamic theory

)(afa Evolution equation:

),...,,,,( aecv a

1 Statics (equilibrium properties)

:)(aSdapdvTdSde

a

S

Te

S

a

a

...

,,1

2 Dynamics

0)()()()( aaaaa fDSDSS

1 + 2 + closed system

S is a Ljapunov function of the equilibrium of the dynamic law

Constructive application:

0

)()),(()(0)()(

DSDS

DSDSDS

S

S

L

aaaLafafa

force current

Classical evolution equations:balances + constitutive assumptions

0)( TTc ee q

TcTe q,

0 TDT Fourier heat conduction

akasla )(' Ginzburg-Landau equation:relaxation + nonlocality

D>0

l>0, k>0

Not so classical evolution equations:balances (?) + constitutive assumptions

Space Time

Strongly nonlocal

Space integrals Memory functionals

Weakly nonlocal

Gradient dependent

constitutive functions

Rate dependent constitutive functions

Relocalized

Current multipliers Internal variables

Nonlocalities:

Restrictions from the Second Law.change of the entropy currentchange of the entropy

Change of the constitutive space

Basic state, constitutive state and constitutive functions:

ee q

– basic state:(wanted field: T(e))

e

)(Cq),( eeC

Heat conduction – Irreversible Thermodynamics

),( ee ))(),(( eTeT T q )())(),((),( eTeTeTee q

Fourier heat conduction:

But: qq LT qqq 21LLT Cattaneo-VernoteGuyer-Krumhansl

– constitutive state:– constitutive functions:

,...),,,,( 2eeeee ???

1)

)(C ),( v C

Local state – Euler equation

0

0

Pv

v

2)

– basic state:– constitutive state:– constitutive function:

Fluid mechanics

Nonlocal extension - Navier-Stokes equation:v

se

p1

),,()()( 2

IP

vIvvP 2))((),( p

But: 22)( IP prKor

),,,( 2 vC),( v

)(CP

Korteweg fluid

fa

a

s

a

sLa

Internal variable

– basic state: aa– constitutive state:

– constitutive function:

A) Local state - relaxation

da

dsLff

da

ds 0

3)

B) Nonlocal extension - Ginzburg-Landau

aaa 2,,

),( aaa

sL

alaslaaasaas )('ˆ,

2)(ˆ),( 2 e.g.

)(Cf

)0)('ˆ( as

Irreversible thermodynamics – traditional approach:

0

J

0ja

sa

– basic state:

– constitutive state:– constitutive functions:

a

Jj ,, sa

),( aa C

Te

s qqJ

Heat conduction: a=e

0

a

js

as

01

2 T

TT

qq

0)(

a

jja

aaa

jaa

Jasssss

s aaa

J=

currents and forces

aLj

s

a

Solution!

Second Law:

aa ja basic balances ,...),( va

– basic state:– constitutive state:– constitutive functions:

a

)C(aj,...),,(C aaa

weakly nonlocalSecond law:

0)()( sCCs J

Constitutive theory: balances are constraints

Method: Liu procedure

(universality)

Liu procedureLEMMA (FARKAS, 1896) Let Ai ≠ 0 be independent vectors in a finite dimensional vector space V, i = 1...n, and

S = {p V∗ | p·Ai ≥ 0, i = 1...n}. The following statements are equivalent for a b V:(i) p·B ≥ 0, for all p S.(ii) There are non-negative real numbers λ1,..., λn such that .

1

n

iiiAB

Vocabulary:elements of V∗ – independent variables,V∗ – the space of independent variables, Inequalities in S – constraints,λi – Lagrange-Farkas multipliers.

Usage:

*

11

,0 VpABpApBp

n

iii

n

iii

B A1

Proof : S is not empty. In fact, for all k, i {1,..., n} there is asuch that pk·Ak = 1 and pk·Ai = 0 if i ≠ k. Evidently pkS for all k.

(ii) (i) if p S.

(i) (ii) Let S0 = {yV∗ | y · Ai = 0, i = 1...n}. Clearly ≠ ∅ S0 S.If y S0 then −y is also in S0, therefore y·B ≥ 0 and −y·B ≥ 0 together.Therefore for all y S0 it is true that y·B = 0.As a consequence B is in the set generated by {Ai}, that is, there are

real numbers λ1,..., λn such that B . These numbers are non-

negative, because with the previously defined pk S,

is valid for all k. QED

*Vp k

011

n

iii

n

iii ApAp

n

iii

1

A

k

n

iiki

n

iiikk

11

0 ApApBp

AFFIN FARKAS: Let Ai ≠ 0 be independent vectors in a finite dimensional vector space V, αi real numbers i = 1...n and

SA = {p V∗ | p · Ai ≥ αi, i = 1...n}. The following statements are equivalent for a B V and a real number :(i) p · B ≥ , for all p SA.(ii) There are non-negative real numbers λ1,..., λn such that

B = and

PROOF: …

.1

n

iii

n

iii

1

A

Vocabulary:Final equality: – Liu equationsFinal inequality: – residual (dissipation) inequality.

LIU’s THEOREM: Let Ai ≠ 0 be independent vectors in a finite dimensional vector space V, αi real numbers i = 1...n and

SL = {p V∗ | p · Ai = αi, i = 1...n}. The following statements are equivalent for a B V and a real number :(i) p · B ≥ , for all p SL.(ii) There are real numbers λ1,..., λn such that

B = and

PROOF: A simple consequence of affine Farkas.

.1

n

iii

n

iii

1

A

Usage:

*

11

1

,0 VpABp

ApBp

n

iii

n

iii

n

iiii

Irreversible thermodynamics – beyond traditional approach:

0

J

0ja

sa

– basic state:

– constitutive state:– constitutive functions:

a

Jj ,, sa

),( aa C

Liu and Müller: validity in every time and space points,derivatives of C are independent:

0::0

::02

2

aJaJaaBp

0ajajaAp

aaaa

aa

ss

aa

),,( 2* aaaV Gen

A) Liu equations:

0a

j

a

J

0aa

a

ss

,,

)(),()('ˆ),(

),(ˆ),(

0 ajaajaaaJ

aaa

as

ss

Te

s

es

qqJ

)(

Spec: Heat conduction: a=e

B) Dissipation inequality:

0'ˆ

a

jjs

s aa

01

2 T

TT

qqA) B)

solution

What is explained:

The origin of Clausius-Duhem inequality: - form of the entropy current - what depends on what

Conditions of applicability!!

- the key is the constitutive space

Logical reduction:

the number of independent physical assumptions!

Mathematician: ok but…Physicist:

no need of such thinking, I am satisfied well and used to my analogiesno need of thermodynamics in general

Engineer:consequences??

Philosopher: …Popper, Lakatos:

excellent, in this way we can refute

Ginzburg-Landau (variational):

dVaasas ))(2

)(ˆ()( 2

))('ˆ( aasla – Variational (!) – Second Law?– ak

aassa )('ˆ

sla a

Weakly nonlocal internal variables

dVaasas ))(2

)(ˆ()( 2

sla a

Ginzburg-Landau (thermodynamic, relocalized)

),,( 2aaa

J),,( sf

Liu procedure (Farkas’s lemma)

)(as

0' fss J

constitutive state space

constitutive functions

fa 0 Js

),( aa J

?

local state

a

saaaa

),(),( BJ

0')(' sfss BB

a

sL

a

sLa 2211

'' 2221 sLsLf B

'' 1211 sLsL B

isotropy

))('( aasla

current multiplier

Ginzburg-Landau (thermodynamic, non relocalizable)

fa

0 Js

),,( 2aaa

J),,( sf

Liu procedure (Farkas’s lemma)

),( aas ),()()( 0 aaCfa

sC

jJ

0

fa

s

a

ss

a

s

a

sLa

state space

constitutive functions 0 fa

Discussion:

– Applications: – heat conduction, one component fluid (Schrödinger-Madelung, …), two component fluids (sand), complex Ginzburg-Landau, … , weakly non-local statistical physics,… – ? Cahn-Hilliard, Korteweg-de Vries, mechanics (hyperstress), …

– Dynamic stability, Ljapunov function???– Universality – independent on the micro-modell– Constructivity – Liu + force-current systems– Variational principles: an explanation

Second Law

References:

Discrete, stability:T. Matolcsi: Ordinary thermodynamics, Publishing House of the Hungarian Academy of Sciences, Budapest, 2005.

Liu procedure:Liu, I-Shih, Method of Lagrange Multipliers for Exploitation of the Entropy Principle, Archive of Rational Mechanics and Analysis, 1972, 46, p131-148.

Weakly nonlocal:Ván, P., Exploiting the Second Law in weakly nonlocal continuum physics, Periodica Polytechnica, Ser. Mechanical Engineering, 2005, 49/1, p79-94, (cond-mat/0210402/ver3).

Ván, P. and Fülöp, T., Weakly nonlocal fluid mechanics - the Schrödinger equation, Proceedings of the Royal Society, London A, 2006, 462, p541-557, (quant-ph/0304062).

Ván, P., Weakly nonlocal continuum theories of granular media: restrictions from the Second Law, International Journal of Solids and Structures, 2004, 41/21, p5921-5927, (cond-mat/0310520).

Thank you for your attention!