1. Weakly nonlocal heat conduction – modeling memory and structure with nonequilibrium thermodynamics Peter Ván HAS, RIPNP, Department of Theoretical Physics –Introduction memory and structure? different heat equations –Memory – Cattaneo-Vernote –Structure – Guyer-Krumhansl –Hierarchy of heat equations - balances –Discussion

2. 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: – Second Law – fundamental balances – objectivity - frame indifference

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

4. Weakly nonlocal memory: inertia T T0T0 Q

6. Non-homogeneous equilibrium: structure s a Single well

7. Double well: two phases s a

8. Stable mixed sructure: twinning in shape memory alloys Initial and boundary conditions!

9. Second Law: basic balances – basic state: – constitutive state: – constitutive functions: Second law: Constitutive theory Method: Liu procedure + … - solving the Liu equations (universality)

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

11. Basic state, constitutive state and constitutive functions: – basic state: Heat conduction – Extended Thermodynamics Heat conduction: Cattaneo-Vernote Guyer-Krumhansl – constitutive state: – constitutive functions: Fourier

12. Weakly nonlocal extended thermodynamics Liu procedure (Farkas’s lemma): constitutive space constitutive functions solution ? local state: state space It is not solvable! Currents and forces?

13. extended (Gyarmati) entropy entropy current (Nyíri) (B – current multiplier) plausible general (dE=TdS ~ q=TJ) concave entropy (stability)

14. gradients Guyer-Krumhansl equation + new terms applications? Liu? Solution, conditions (e.g. L 11 p.d.)

15. Weakly nonlocal extended thermodynamics (again) constitutive space constitutive functions state space Specific questions: balance form Why ? hierarchy Closure ? locality Why ?

16. Weakly nonlocal extended thermodynamics (again) constitutive space constitutive functions state space First order nonlocality Liu procedure (Farkas’s lemma): Liu equations

17. Liu’s theorem: Usage: Conditions: Consequences:

18. Balance form evolution + local s, H: Liu equations: Dissipation inequality:

19. “New” independent variables! potential structure no dissipation

20. Weakly nonlocal extended thermodynamics (again) constitutive space constitutive functions state space Second order nonlocality + local state: s(e,q) Liu procedure (Farkas’s lemma): Liu equations

21. current multiplier extended entropy Once more: Almost balance: Closed (trivial)

22. Discussion: – Kinetic – phenomenological – Universality – independent on the micro-modell – Constructivity – Liu + force-current systems – Origin of balances? – Closure – C=(weakly nonlocal in time)? Second Law

23. References: General: Gyarmati, I., The wave approach of thermodynamics and some problems of non-linear theories, Journal of Non-Equilibrium Thermodynamics, 1977, 2, p233-260. Müller, I. and Ruggeri, T.: Rational Extended Thermodynamics, Springer Verlag, 1998, Springer Tracts in Natural Philosophy V 37, New York-etc. Jou, D., Casas-Vázquez, J. and Lebon, G., Extended Irreversible Thermodynamics, Springer Verlag, 2001, Berlin-etc., 3rd, revised edition. Heat conduction: 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). Ciancio, V., Cimmelli, V. A. and Ván, P., On the evolution of higher order fluxes in non- equilibrium thermodynamics, 2006, (cond-mat/0407530).

24. Thank you for your attention!