Introduction Second Law Weak nonlocality Liu procedure Classical irreversible thermodynamics

1. Weakly nonlocal continuum physics – the role of the Second LawPeterVán HAS, RIPNP, Department of Theoretical Physics • Introduction • Second Law • Weak nonlocality • Liu procedure • Classical irreversible thermodynamics • Ginzburg-Landau equation • Discussion

2. Nonequilibrium thermodynamics science of temperature Thermodynamics science of macroscopic energy changes Thermodynamics • general framework of any • Thermodynamics (?) macroscopic • continuum • theories • General framework: • Second Law • fundamental balances • objectivity - frame indifference reversibility – special limit

3. Thermo-Dynamic theory Evolution equation: 1 Statics (equilibrium properties) 2 Dynamics

4. 1 + 2 + closed system S is a Ljapunov functionof the equilibrium of the dynamic law Constructive application: force current

5. Classical evolution equations: balances + constitutive assumptions Fourier heat conduction D>0 Not so classical evolution equations: balances (?) + constitutive assumptions Ginzburg-Landau equation: relaxation +nonlocality l>0, k>0

6. Nonlocalities: Restrictions from the Second Law. change of the entropy current change of the entropy Change of the constitutive space

7. Basic state, constitutive state and constitutive functions: Heat conduction – Irreversible Thermodynamics 1) • basic state: • (wanted field:T(e)) • constitutive state: • constitutive functions: Fourier heat conduction: But: Guyer-Krumhansl Cattaneo-Vernote ???

8. Fluid mechanics 2) • basic state: • constitutive state: • constitutive function: Local state – Euler equation Nonlocal extension - Navier-Stokes equation: But: Korteweg fluid

9. Internal variable 3) • basic state: • constitutive state: • constitutive function: A) Local state - relaxation B) Nonlocal extension - Ginzburg-Landau e.g.

10. Irreversible thermodynamics – traditional approach: • basic state: • constitutive state: • constitutive functions: J= Solution! currents andforces Heat conduction: a=e

11. weakly nonlocal Second Law: basic balances • basic state: • constitutive state: • constitutive functions: Second law: (universality) Constitutive theory: balances are constraints Method: Liu procedure

12. Liu procedure LEMMA (FARKAS, 1896) Let Ai≠0 be independent vectors in a finite dimensionalvector space V, i = 1...n, and S = {p V∗ | p·Ai≥ 0, i = 1...n}. Thefollowing 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 Vocabulary: elements of V∗ – independent variables, V∗ – the space of independent variables, Inequalities in S – constraints, λi – Lagrange-Farkas multipliers.

13. Usage: A1 B

14. Proof : S is not empty. In fact, for all k, i {1,..., n} there is a such that pk·Ak= 1 and pk·Ai= 0 if i ≠ k. Evidently pkS for all k. (ii) (i) if pS. (i)  (ii) Let S0 = {yV∗ | 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,..., λnsuch that B. These numbers are non- negative, because with the previously defined pkS, is valid for all k. QED

15. AFFIN FARKAS: Let Ai≠0 be independent vectors in a finite dimensionalvector space V, αi real numbers i = 1...n and SA= {p V∗ | p · Ai≥ αi, i = 1...n}. Thefollowing 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: … Vocabulary: Final equality: – Liu equations Final inequality: – residual (dissipation) inequality.

16. LIU’s THEOREM: Let Ai≠0 be independent vectors in a finite dimensionalvector space V, αi real numbers i = 1...n and SL= {p V∗ | p · Ai= αi, i = 1...n}. Thefollowing statements are equivalent for a B Vand 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. Usage:

17. Irreversible thermodynamics – beyond traditional approach: • basic state: • constitutive state: • constitutive functions: Liu and Müller: validity in every time and space points, derivatives of C are independent:

18. A) Liu equations: solution B) Dissipation inequality: Spec: Heat conduction: a=e A) B)

19. 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 physicalassumptions! Mathematician: ok but… Physicist: no need of such thinking, I am satisfied well and used to my analogies no need of thermodynamics in general Engineer: consequences?? Philosopher: … Popper, Lakatos: excellent, in this way we can refute

20. Weakly nonlocal internal variables Ginzburg-Landau (variational): • Variational (!) • Second Law?

21. Ginzburg-Landau (thermodynamic, relocalized) constitutive state space constitutive functions local state Liu procedure (Farkas’s lemma) ?

22. current multiplier isotropy

23. Ginzburg-Landau (thermodynamic, non relocalizable) state space constitutive functions Liu procedure (Farkas’s lemma)

24. 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

25. 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 SecondLaw in weakly nonlocal continuum physics, PeriodicaPolytechnica, Ser. Mechanical Engineering, 2005, 49/1, p79-94, (cond-mat/0210402/ver3). Ván, P. and Fülöp, T., Weakly nonlocal fluid mechanics - the Schrödingerequation, Proceedings of the Royal Society,London A, 2006, 462, p541-557, (quant-ph/0304062). Ván, P., Weakly nonlocalcontinuum theories of granular media: restrictions from theSecond Law, International Journal of Solids andStructures, 2004, 41/21, p5921-5927, (cond-mat/0310520).

26. Thank you for your attention!