 Phases

2.  Phases Philippe CHOMAZ - GANIL Thermodynamic, Equilibrium Statistical ensemble Observation space Information & entropy Gibbs Equilibria Example: Mean-field Discussion What is temperature? What is equilibrium? Ensemble inequivalence Time dependent equilibria Nuclear matter Isospin dependent EOS Phase transition Spinodal decomposition Neutron* & Supernovae Phase transi. finite system Zero of partition sum Bimodalities C negative 1

4. Absolute necessity of the second principle R. Balian « Statistical mechanics » 52

5. p q Absolute necessity of the second principle R. Balian « Statistical mechanics » • First principle: energy conservation • Time independent laws (symmetry) => E conserved • Classical : Point in phase space • Quantum : Vector of Hilbert space (q)q| q 52

6. p q Absolute necessity of the second principle R. Balian « Statistical mechanics » • First principle: energy conservation • Time independent laws (symmetry) => E conserved • Classical : Point in phase space • Quantum : Vector of Hilbert space (q)q| t0 q t0 t0 t0 52

7. p q Absolute necessity of the second principle R. Balian « Statistical mechanics » • First principle: energy conservation • Time independent laws (symmetry) => E conserved • Classical : Point in phase space • Quantum : Vector of Hilbert space (q)q| t0 q t0 t0 t0 52

8. p q Absolute necessity of the second principle R. Balian « Statistical mechanics » • Initial condition => infinite information needed • Infinite accuracy needed (Chaos) • Classical : 6.N coordinates Point in phase space • Quantum : 2.∞ coordinates Vector of Hilbert space (q)q| q 52

9. Absolute necessity of the second principle R. Balian « Statistical mechanics » • Initial condition => infinite information needed • Infinite accuracy needed (Chaos) • Classical : 6.N coordinates Point in phase space • Quantum : 2.∞ coordinates Vector of Hilbert space • Degree of freedom => infinite information needed • Our ignorance of initial comdition should be taken into account to make the theory meaningful 52

10. <p> p <q> q Classical Chaos <= Quantum ∞ D. freedom • Classical : 6.N coordinates Chaos • Quantum : 2.∞ coordinates Projection <p> <p2> <q> <qp> <q2> • Our ignorance of initial comdition should be taken into account to make the theory meaningful 52

12. -I- ThermodynamicsInformation theoryStatistical physics 2

14. A-Thermo & Statistical ensembles R. Balian « Statistical mechanics » 52

15. } { , , , … A-Ensembles R. Balian « Statistical mechanics » • Ensemble of events / partitions / replicas : • State • Classical : Point in phase space • Ensemble = {states+occurrence probability } => Phase space density • Quantum : Vector of Hilbert space => Density Matrix 52

16. One macroscopic system is an ensemble Thermodynamics : infinite system One ∞ system = ensemble of ∞ sub-systems

17. ≠ Surfaces Cannot be neglected ≠ A single microscopic system ≠ ensemble Finite system Cannot be cut in sub-systems

18. A single microscopic system ≠ ensemble Thermodynamics & statistical physics do not apply to a single realization of a finite system Cannot be cut in sub-systems

19. Cannot be cut in sub-systems Thermo describe several realizations One small system in time => statistical ensemble

20. Thermo describe several realizations One small system in time => statistical ensemble Many events

22. B-Observation space R. Balian « Statistical mechanics » 52

23. R. Balian « Statistical mechanics » • State Classical : Phase space Density Quantum : Density Matrix 52

24. B-Observation space R. Balian « Statistical mechanics » • State Classical : Phase space Density Quantum : Density Matrix • Observables Phase space functions Operators (Matrices) • Observation 52

25. Observation 52

26. <Â3> <Â2> ^ D <Â1> Geometrical interpretation of observation • Scalar product in Observable space • Observation • Observation 52

27. <Â3> <Â2> ^ D <Â1> Geometrical interpretation of observation • Scalar product in Observable space • Observation HUGE (Infinite) space Classical => N*6 ; Quantum => N*∞ 52

29. ^ ^ ^ ^ ^ ^ y z x Non locality = p-dependence Wigner Transform Basis of operator space (1 particle) • Spatial • Hilbert basis =>{|r>} (or {|p>}) : j(r) = <r|j> • Operators =><r|O|r’> (or <p|O|p’>) =>O = f(r,p) =∑oijklmn xiyjzkplpnpm ^ ^ ^ ^ ^ 8

30. ^ ^ ^ ^ ^ ^ y z x Basis of operator space (1 particle) • Spatial • Hilbert basis =>{|r>} (or {|p>}) : j(r) = <r|j> • Operators =><r|O|r’> (or <p|O|p’>) =>O = f(r,p) =∑oijklmn xiyjzkplpnpm ^ ^ ^ ^ ^ 8

31. <xipl> ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ y y z z <p> x x ^ D ^ ^ <x> xiyjzkplpnpm ^ ^ Basis of operator space (1 particle) • Spatial • Hilbert basis =>{|r>} (or {|p>}) : j(r) = <r|j> • Operators =><r|O|r’> (or <p|O|p’>) =>O = f(r,p) =∑oijklmn xiyjzkplpnpm => Infinite basis ^ ^ ^ ^ ^ Infinite space 8

32. <xipl> <p> ^ D ^ ^ <x> ^ ^ Require the treatment of our ignorance • Initial condition cannot be known • The dynamics cannot be followed • Impossible to know everything • Only part of the information is relevant Infinite space 8

34. C- Time evolution R. Balian « Statistical mechanics » 52

35. R. Balian « Statistical mechanics » • State Classical : Phase space Density Quantum : Density Matrix 52

36. C- Time evolution R. Balian « Statistical mechanics » • State Classical : Phase space Density Quantum : Density Matrix • Dynamics Hamilton Schrödinger Liouville-von Neumann Liouville 52

37. C- Time evolution R. Balian « Statistical mechanics » • Observation Classical : Quantum : • Dynamics Heisenberg (Ehrenfest) • State Liouville-von Neumann Liouville 52

39. C- Information and Entropy R. Balian « Statistical mechanics » 52

40. C- Information and Entropy R. Balian « Statistical mechanics » • Shannon information of probability distribution p(n) • Measure the Information • Max when we know everything • Min when we know nothing • Decrease with our ignorance • Concavity • Additivity • Entropy

42. D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » 52

43. D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » • Gibbs equilibria are minimum bias distributions => distribution maximizing the entropy • Example Nothing known => States equiprobable => Microcanonical

44. D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » R. Balian « Statistical mechanics » • Statistical ensemble: • Equilibrium = Max S: • Constraints (<H>, <N>): (Variational principle) ^ ^ • Lagrange multipliers Boltzman distribution Partition sum Equation of state 52

45. D- Equilibrium et minimum bias (max S) Equilibrium ensembles R. Balian « Statistical mechanics » R. Balian « Statistical mechanics » • Statistical ensemble: • Equilibrium = Max S: • Constraints (<H>, <N>): (Minimum information) ^ ^ • Lagrange multipliers Boltzman distribution Partition sum Equation of state • Equilibrium entropy 52

46. D- Equilibrium et minimum bias (max S) B-Thermodynamics R. Balian « Statistical mechanics » R. Balian « Statistical mechanics » • Statistical ensemble: • Equilibrium = Max S: • Constraints (<H>, <N>): (Variational principle) ^ ^ • Lagrange multipliers Boltzman distribution Partition sum Equation of state • Equilibrium entropy 53

47. Example: mean field R. Balian « Statistical mechanics » • Trial state: • Sc functional of r: • Max constrained S: =>like in a mean field =>Equilibrium O=W =>Fermi-dirac statistic =>Mean field entropy (Independent particles) ^ (Variational principle) • Best approximation Sc: =>Best approximation logZ: 54

49. -II- DiscussionTemperature Equilibra 2