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Introduction Stability and thermodynamics Gases Discussion

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. Mechani cs:.

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Introduction Stability and thermodynamics Gases Discussion

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

  2. Mechanics: Statics Dynamics Point-massContinuum Thermodynamics: Equilibrium Non-equilibrium Phenomenological Statistical Discrete (homogeneous) Continuum

  3. Discrete systems – equilibrium thermodynamics Equilibrium thermodynamics 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 = drS + diS és diS0 • ….

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

  5. What is? Equilibrium of (*): 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).

  6. What is? Instead of proof: x1 x2

  7. What is? T0, p0 T, p Dynamics without differential equation? A) entropy q as a potential. B) ? ‘dynamic law?’

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

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

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

  11. Quasistatic processes of a Van der Waals gas:

  12. v0 T0 p0 Pitchfork bifurcation of a Van der Waals gas (bifurcation diagram)

  13. What is? T0, p0 T, p Second order equation – internal variables q State space: Non-equilibrium state functions: e.g.

  14. What is? Stability structure: Ljapunov function Viscous-damping: Entropy of the body: dS = drS + diS és diS0

  15. Irreversible processes of a Van der Waals gas:

  16. Movie-like:

  17. What is  for what? 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.

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

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