1. What is thermodynamics and what is it for? I. Equilibrium and non-equilibrium in discrete systems Peter Ván HAS, RIPNP, Department of Theoretical Physics –Introduction –Stability and thermodynamics –Gases –Discussion Centre of Nonlinear Studies, Tallinn, Estonia, 29/5/2006.

2. Thermodynamics: Statics Dynamics Point-mass Continuum Mechanics: EquilibriumNon-equilibrium PhenomenologicalStatistical Discrete (homogeneous) Continuum

3. Equilibrium - non-equilibrium? –Mechanics – thermodynamics Differential equations? –Equilibrium - time independent –Quasistatic processes – time? (Zeno) –Irreversible processes – something more (internal variables) II. law? –Heat can flow from a hotter body to the colder. (Clausius) –There is no perpetuum mobile of the second kind. (Planck) –In a closed system, in case of spontaneous processes the entropy increases. –dS = d r S + d i S és d i S  0 –…. Discrete systems – equilibrium thermodynamics Equilibrium thermodynamics

4. – S entropy – de = q+w = Tds – pdv Gibbs relation – There is a tendency to equilibrium – Thermodynamic stability What is? Basic ingredients:

5. L: E  is a Ljapunov function of the equilibrium, if: i) L has a strict maximum at, ii), the derivative of L along the differential equation has a strict minimum at. Theorem: If there is a Ljapunov function of, then is asymptotically stable (stable and attractive). Equilibrium of (*): What is?

6. Instead of proof: x2x2 x1x1 What is?

7. Dynamics without differential equation? ? ‘dynamic law?’ A) entropy B) T, p T 0, p 0 as a potential. q What is?

8. Stability structure: Ljapunov function i) ii) convex – thermodynamic stability Direction of heat:

9. convex E.g. Dynamic Law Dynamic material functions (heat exchange, …) Static materialfunctions (ideal gas)

10. Thermodynamic theory in general Dynamic Law: 1 Statics (properties of equilibrium): existence of entropy 2 Dynamics (properties of interactions) : increasing entropy Stability structureDynamic structure What for?

11. Quasistatic processes of a Van der Waals gas:

12. Pitchfork bifurcation of a Van der Waals gas (bifurcation diagram) v0v0 T0T0 p0p0

13. Second order equation – internal variables Non-equilibrium state functions: T, p T 0, p 0 State space: q What is? e.g.

14. Stability structure: Ljapunov function Viscous-damping: What is? Entropy of the body: dS = d r S + d i S és d i S  0

15. Irreversible processes of a Van der Waals gas:

16. Movie-like:

17. Conclusions –A thermodynamic structure is a stability structure Time dependent discrete systems! Equilibrium – quasistatic – irreversible –Completing the structure: theory construction Static: consistency, thermic  caloric Dynamics: Onsager reciprocity, constitutive functions, Constructive! – Stability of the theory: stability of the calculations. Robust numerical codes: numerical viscosity –Discrete – continuum: the same principles. What is  for what?

18. Thanks: To the Hungarian thermodynamic tradition: Julius Farkas, Imre Fényes, István Gyarmati,. Joe Verhás, Tamás Matolcsi Thermodynamic Division of the Hungarian Physical Society,