– Introduction • Second Law

• Weak nonlocality

– Ginzburg-Landau equation

– Schrödinger-Madelung equation

– Digression: Stability and statistical physics

– Discussion

Weakly nonlocal nonequilibrium thermodynamics –fluids and beyond

Peter Ván BCPL, University of Bergen, Bergen and

RMKI, Department of Theoretical Physics, Budapest

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

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)

fa

a

s

a

sLa

Internal variable

– basic state: aa– constitutive state:

– constitutive function:

A) Local state - relaxation

0 fda

ds

da

dsLa

2)

B) Nonlocal extension - Ginzburg-Landau

aaa 2,,

),( aaa

sL

alaslaaasaas )('ˆ,

2)(ˆ),( 2 e.g.

)(Cf

)0)('ˆ( as

)(C ),( v C

Local state – Euler equation

0

0

Pv

v

3)

– 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

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!

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

1

2

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

constitutive state space

constitutive functions 0 fa

),,( aaaC xxx

),(

),(0

;;

33 aajfsJfJ

aass

ss

xaxx

xa

aa

x

xx

x

Liu equations:

0)(

fa

s

a

s

xxs

0)()(

)()()(

2211

33321

fafJafJ

fJasasasa

xxxxx

xxxxtxxtxt

)()( 321

321321

afafafafa

aJaJaJasasas

xxxxxxtxt

xxxxxxxxxtxxtxt

))()(())()(()()( CfaCCfaCCJCs xtxtxxt

constitutive state space

Korteweg fluids (weakly nonlocal in density, second grade)

),,( v C ),,,( v wnlC

)(),(),( CCCs PJ

Liu procedure (Farkas’s lemma):

constitutive state

constitutive functions

0 v

0)()( CCs J0Pv )C(

...J)(ess ),(),( ess

),( v basic state

0:s2

ss2

1 22

s

vIP

rv PPP

reversible pressurerP

Potential form: nlr U P

)()( eenl ssU Euler-Lagrange form

Variational origin

Spec.: Schrödinger-Madelung fluid2

22),(

SchM

SchMs

2

8

1 2IP rSchM

(Fisher entropy)

Potential form: Qr U P

Bernoulli equation

Schrödinger equation

v ie

R1: Thermodynamics = theory of material stability

In quantum fluids:– There is a family of equilibrium (stationary) solutions.

0v .constEUU SchM

– There is a thermodynamic Ljapunov function:

dVEUL

22

22

1

2),(

v

v

semidefinite in a gradient (Soboljev ?) space

2

xD)(xU

2

Mov1.exe

– Isotropy– Extensivity (mean, density)

– Additivity

Entropy is unique under physically reasonable conditions.

R2: Weakly nonlocal statistical physics:

Boltzmann-Gibbs-Shannon

)()( ss

)()()( 2121 sss

ln)( ks

))(,(),( 2 ss

),(),())(,( 22112121 ssDs

2

22 )(

ln))(,(

ks

-2 -1 1 2x

0.2

0.4

0.6

0.8

1

1.2

R

18

,12

),5,2,5.1,3.1,2.1,1.1,1(4 111

mkk

k

k

k

Discussion:

– Applications: – heat conduction (Guyer-Krumhansl), Ginzburg-Landau, Cahn-Hilliard, one component fluid (Schrödinger-Madelung, etc.), two component fluids (gradient phase trasitions), … , weakly nonlocal statistical physics,… – ? Korteweg-de Vries, mechanics (hyperstress), …

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

Thermodynamics – theory of material stability

References:

1. 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).

2. 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).

3. P. Ván and T. Fülöp. Stability of stationary solutions of the Schrödinger-Langevin equation. Physics Letters A, 323(5-6):374(381), 2004. (quant-ph/0304190)

4. 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).

5. Cimmelli, V. A. and Ván, P., The effects of nonlocality on the evolution of higher order fluxes in non-equilibrium thermodynamics, Journal of Mathematical Physics, 2005, 46, p112901, (cond-mat/0409254).

6. V. Ciancio, V. A. Cimmelli, and P. Ván. On the evolution of higher order fluxes in non-equilibrium thermodynamics. Mathematical and Computer Modelling, 45:126(136), 2007. (cond-mat/0407530).

7. P. Ván. Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond. Physica A, 365:28(33), 2006. (cond-mat/0409255)

8. P. Ván, A. Berezovski, and Engelbrecht J. Internal variables and dynamic degrees of freedom. 2006. (cond-mat/0612491)

Thank you for your attention!