2. What is thermodynamics and what is it for? II. Continuum physics – constitutive theory Peter Ván HAS, RIPNP, Department of Theoretical Physics –Introduction –Constitutive space and constitutive functions –Classical irreversible thermodynamics –Weakly non-local extensions Internal variables, heat conduction and fluids –Discussion Centre of Nonlinear Studies, Tallinn, Estonia, 19/6/2006.

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

4. 1 + 2 + closed system S is a Ljapunov function of the equilibrium of the dynamic law Constructive application: forcecurrent

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

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

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

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

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

10. Second Law: basic balances – basic state: – constitutive state: – constitutive functions: weakly nonlocal Second law: Constitutive theory Method: Liu procedure (universality) (and more)

11. Irreversible thermodynamics: – basic state: – constitutive state: – constitutive functions: primary!! Liu procedure (Farkas lemma): A) Liu equations: Heat conduction: a=e B) Dissipation inequality:

12. 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 analogies no need of thermodynamics in general Engineer: consequences?? Philosopher: … Popper, Lakatos: excellent, in this way we can refute

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

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

15. isotropy current multiplier

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

17. Weakly nonlocal extended thermodynamics Liu procedure (Farkas’s lemma): constitutive space constitutive functions solution ? local state: state space

18. extended (Gyarmati) entropy entropy current (Nyíri) (B – current multiplier) gradient Guyer-Krumhansl equation

19. Korteweg fluids ( weakly nonlocal in density, second grade) Liu procedure (Farkas’s lemma): constitutive state constitutive functions basic state

20. reversible pressure Potential form: Euler-Lagrange form Variational origin

21. Schrödinger-Madelung fluid (Fisher entropy) Bernoulli equation Schrödinger equation

22. Thermodynamics = theory of material stability Ideas: –Phase transitions in gradient systems? In quantum fluids: –There is a family of equilibrium (stationary) solutions. –There is a thermodynamic Ljapunov function: semidefinite in a gradient (Soboljev ?) space

24. Conclusions -Dynamic stability, Ljapunov function??? -Universality – independent on the micro-modell -Constructivity – Liu + force-current systems -Variational principles: an explanation Second Law Problems, perspectives: objectivity (material frame indifference): mechanics (hyperstress and strain)! electrodynamics (special relativity) But: heat conduction, two component fluids (sand), Cahn-Hilliard, complex Ginzburg- Landau, Korteweg-de Vries, …., weakly non-local statistical physics, …

25. Thank you for your attention!