Nuclear Phases India 2006 80 Thermodynamic Equilibrium Phases

Nuclear Phases, India, 2006 : 80

Thermodynamic, Equilibrium ������� Phases Philippe CHOMAZ - GANIL Discussion Isospin dependent EOS Phase transition Spinodal decomposition Neutron* & Supernovae Phase transi. finite system Nuclear Phases, India, 2006 What is temperature? What is equilibrium? Ensemble inequivalence Time dependent equilibria Nuclear matter Statistical ensemble Observation space Information & entropy Gibbs Equilibria Example: Mean-field 1: 80 Zero of partition sum Bimodalities C negative

Nuclear Phases, India, 2006 : 80

Absolute necessity of the second principle R. Balian « Statistical mechanics » Nuclear Phases, India, 2006 52 : 80

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 Vector of Hilbert space p (q) q| q Nuclear Phases, India, 2006 Quantum : q 52 : 80

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 p (q) q| t 0 Nuclear Phases, India, 2006 q t 0 52 : 80 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 p (q) q| t 0 Nuclear Phases, India, 2006 q t 0 52 : 80 q

Absolute necessity of the second principle R. Balian « Statistical mechanics » Initial condition => infinite information needed Infinite accuracy needed (Chaos) Classical : Quantum : 6. N coordinates 2. ∞ coordinates Point in phase space Vector of Hilbert space p (q) q| q Nuclear Phases, India, 2006 q 52 : 80

Absolute necessity of the second principle R. Balian « Statistical mechanics » Initial condition => infinite information needed Infinite accuracy needed (Chaos) Classical : Quantum : 6. N coordinates 2. ∞ coordinates Point in phase space Vector of Hilbert space Degree of freedom => infinite information needed Our ignorance of initial comdition should be taken into account to make theory meaningful Nuclear Phases, India, 2006 52 : 80

Classical Chaos <= Quantum ∞ D. freedom Classical : 6. N coordinates 2. ∞ coordinates Chaos Projection p <p> <p 2> <qp> <q 2> q Quantum : <q> Our ignorance of initial comdition should be taken into account to make theory meaningful Nuclear Phases, India, 2006 52 : 80

Nuclear Phases, India, 2006 : 80

-IThermodynamics Information theory Statistical physics Nuclear Phases, India, 2006 2: 80

Nuclear Phases, India, 2006 : 80

A-Thermo & Statistical ensembles R. Balian « Statistical mechanics » Nuclear Phases, India, 2006 52 : 80

A-Ensembles R. Balian « Statistical mechanics » Ensemble of events / partitions / replicas : { , , , … State Classical : Point in phase space } Quantum : Vector of Hilbert space Ensemble = {states+occurrence probability => Density Matrix => Phase space density Nuclear Phases, India, 2006 52 : 80 }

One macroscopic system is an ensemble Thermodynamics : infinite system One ∞ Nuclear Phases, India, 2006 system = ensemble of : 80 ∞ sub-systems

A single microscopic system ≠ Finite system Cannot be cut in sub-systems Nuclear Phases, India, 2006 ≠ ensemble : 80 ≠ Surfaces Cannot be neglected

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 Nuclear Phases, India, 2006 : 80

Thermo describe several realizations One small system in time => statistical ensemble Cannot be cut in sub-systems Nuclear Phases, India, 2006 : 80

Thermo describe several realizations One small system in time => statistical ensemble Many events Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80

B-Observation space Nuclear Phases, India, 2006 R. Balian « Statistical mechanics » 52 : 80

R. Balian « Statistical mechanics » State Classical : Quantum : Phase space Density Matrix Nuclear Phases, India, 2006 52 : 80

B-Observation space R. Balian « Statistical mechanics » State Classical : Quantum : Phase space Density Matrix Observables Operators (Matrices) Phase space functions Observation Nuclear Phases, India, 2006 52 : 80

Observation Nuclear Phases, India, 2006 52 : 80

Geometrical interpretation of observation Scalar product in Observable space < 3> Observation ^ D < 2> < 1> Observation Nuclear Phases, India, 2006 52 : 80

Geometrical interpretation of observation Scalar product in Observable space < 3> Observation ^ D < 2> < 1> HUGE (Infinite) space Classical => N*6 ; Quantum => N*∞ Nuclear Phases, India, 2006 52 : 80

Nuclear Phases, India, 2006 : 80

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 iy^jz^kp ^lp^np^m ^ => O = f(r, p) x ijklmn x y z Non locality = p-dependence Wigner Transform Nuclear Phases, India, 2006 8: 80

Basis of operator space (1 particle) Spatial Hilbert basis Operators Nuclear Phases, India, 2006 => {|r>} (or {|p>}) : j(r) = <r|j> ^ ^ => <r|O|r’> (or <p|O|p’>) ^ ^ ^ =∑o iy^jz^kp ^lp^np^m ^ => O = f(r, p) x ijklmn x y z 8: 80

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 iy^jz^kp ^ = f(r, p) ^lp^np^m ^ => O x ijklmn x y z => Infinite basis ^ip^l> <x ^ D Infinite space ^ <p> ^ <x> Nuclear Phases, India, 2006 x^iy^jz^kp^lxp^nyp^mz 8: 80

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 ^ip^l> <x ^ D Infinite space ^ <p> ^ <x> Nuclear Phases, India, 2006 8: 80

Nuclear Phases, India, 2006 : 80

C- Time evolution Nuclear Phases, India, 2006 R. Balian « Statistical mechanics » 52 : 80

R. Balian « Statistical mechanics » State Classical : Quantum : Phase space Density Matrix Nuclear Phases, India, 2006 52 : 80

C- Time evolution R. Balian « Statistical mechanics » State Quantum : Density Matrix Classical : Phase space Density Dynamics Hamilton Schrödinger Liouville-von Neumann Nuclear Phases, India, 2006 52 : 80

C- Time evolution R. Balian « Statistical mechanics » Observation Quantum : Classical : Dynamics Heisenberg (Ehrenfest) State Liouville Nuclear Phases, India, 2006 Liouville-von Neumann 52 : 80

Nuclear Phases, India, 2006 : 80

C- Information and Entropy Nuclear Phases, India, 2006 52 : 80 R. Balian « Statistical mechanics »

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 Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80

D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » Nuclear Phases, India, 2006 52 : 80

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 Nuclear Phases, India, 2006 : 80

D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » Statistical ensemble: Equilibrium = Max S: Lagrange multipliers Boltzman distribution Partition sum Equation of state (Variational principle) ^ <N>) ^ : Constraints (<H>, Nuclear Phases, India, 2006 52 : 80

Equilibrium D- Equilibrium ensembles et minimum bias (max S) R. Balian « Statistical mechanics » Statistical ensemble: Equilibrium = Max S: Lagrange multipliers Boltzman distribution Partition sum Equation of state Equilibrium entropy (Minimum information) ^ <N>) ^ : Constraints (<H>, Nuclear Phases, India, 2006 52 : 80

B-Thermodynamics D- Equilibrium et minimum bias (max S) R. Balian « Statistical mechanics » Statistical ensemble: Equilibrium = Max S: Lagrange multipliers Boltzman distribution Partition sum Equation of state Equilibrium entropy (Variational principle) ^ <N>) ^ : Constraints (<H>, Nuclear Phases, India, 2006 53 : 80

Example: mean field Trial state: ^ Sc functional of r: Max constrained S: R. Balian « Statistical mechanics » (Independent particles) (Variational principle) =>like in a mean field =>Equilibrium O=W =>Fermi-dirac statistic =>Mean field entropy Best approximation Sc: =>Best approximation log. Z: Nuclear Phases, India, 2006 54 : 80

Nuclear Phases, India, 2006 : 80

-IIDiscussion Temperature Equilibra Nuclear Phases, India, 2006 2: 80

Nuclear Phases, India, 2006 : 80

A- What is temperature ? T W equiprobable microstates Entropy = disorder L. Boltzmann (of the ensemble) T is the entropy increase R. Clausius Nuclear Phases, India, 2006 : 80

A- What is temperature ? The microcanonical temperature S =T-1 E S = k log. W Nuclear Phases, India, 2006 : 80

A- What is temperature ? It is what thermometers measure. E = Ethermometer + Esystem The microcanonical temperature S =T-1 E S = k log. W Nuclear Phases, India, 2006 : 80

A- What is temperature ? It is what thermometers measure. E = Ethermometer + Esystem Distribution of microstates The microcanonical temperature S =T-1 E S = k log. W Nuclear Phases, India, 2006 : 80

A- What is temperature ? It is what thermometers measure. E = Eth + Esys P(Eth) Distribution of microstates E 0 Eth 0 The microcanonical temperature S =T-1 E S = k log. W Nuclear Phases, India, 2006 : 80

A- What is temperature ? It is what thermometers measure. E = Eth + Esys P(Eth) Equiprobable microstates 0 P(Eth) Wth (Eth)* Wsys(E-Eth) max P => log. Wth - log. Wsys =0 max P => Sth - Ssys =0 E Eth 0 Most probable partition: Tth = sys The microcanonical temperature S =T-1 E S = k log. W Nuclear Phases, India, 2006 : 80

Realization of a cononical ensemble The thermometers is canonically distributed E = Eth + Esys P(Eth) Equiprobable microstates P(Eth) Wth (Eth)* Wsys(E-Eth) E 0 0 Eth Small thermometer (Eth small) Ssys (E-Eth)= Ssys (E)- Eth/T Ssys =log. Wsys Boltzmann Nuclear Phases, India, 2006 P(Eth) Wth (Eth)* exp(-Eth/T) : 80

Nuclear Phases, India, 2006 : 80

B- What are the various equilibria? Nuclear Phases, India, 2006 : 80

B- What are the various equilibria? R. Balian « Statistical mechanics » Macroscopic One realization (event) can be an equilibrium One ∞ system = ∞ ensemble of ∞ sub-systems Nuclear Phases, India, 2006 : 80

B- What are the various equilibria? R. Balian « Statistical mechanics » Macroscopic One realization (event) can be an equilibrium One ∞ system = ∞ ensemble of ∞ sub-systems Microscopic Ensemble of replicas needed One realization (event) cannot be an equilibrium Gibbs: Equilibrium = maximum entropy Average over time Average over events Average over replicas ! Nuclear Phases, India, 2006 if ergodic if chaotic/stochastic if minimum info “Ergodic” some times used instead of “uniform population of phase space” : 80

B- What are the various equilibria? R. Balian « Statistical mechanics » ∞ time average = phase space average Ergodic => <≠ statistics Only conserved quantities (E, J, P …) Mixing p p t q -> q Unknown initial conditions p Ergodic (Bound systems only) ! ! p Stochastic q Unknown dynamics Not only conserved statistical variables Complex / min info q Few relevant observations <Al> => state variables (not only conserved) Many irrelevant degree of freedom Nuclear Phases, India, 2006 : 80 p p q q

Validity conditions R. Balian « Statistical mechanics » Bound systems only For time averages only Should be demonstrated (difficult) Only conserved quantities (E, J, P …) Mixing Complex / min info p Ergodic Stochastic For ensemble of events Should be demonstrated (difficult) Comparison with models Consistency checks (e. g. T 1=T 2, s. A 2=-∂l log Z) Independence upon history How far are we from equilibrium? Nuclear Phases, India, 2006 : 80 q p q

Information theory for finite system R. Balian « Statistical mechanics » Statistical ensemble: Shannon information: Information = observations: Min. bias state: min I under constraints Lagrange multipliers => Boltzman probability Partition sum Constraints = EOS Nuclear Phases, India, 2006 : 80

Many different ensembles Constraints E Microcanonical Conserved <E> V Isochore <r 3> Isobare <Q 2> Deformed <p. r> Expanding <A> Grand <L> Rotating . . . Others quantities Sorting Boundaries ! Boundaries = spatial constraints, ex: <V>=<r 3> Boundary = ∞ information Microcanonical undefined => => isobar ensemble Valid also for open systems (extension <r 2>) Nuclear Phases, India, 2006 : 80

C-Finite systems: ensemble inequivalence Nuclear Phases, India, 2006 : 80

C-Finite systems: ensemble inequivalence R. Balian « Statistical mechanics » Géneral ref. : Phase trans. : Nuclear Phases, India, 2006 : 80

C-Finite systems: ensemble inequivalence R. Balian « Statistical mechanics » Microcanonical ensemble: Shannon = Boltzmann: Temperature (EOS): Canonical ensemble: Partition sum = Laplace tr. : Caloric curve (EOS) Canonical Sc = Legendre tr. : But canonical Sc(<E>) ≠ microcanonical S(E) => Canonical EOS ≠ microcanonical EOS Nuclear Phases, India, 2006 : 80 !

Inequivalence Energy Distribution Canonical Energy dist. Exact link microcan. entropy Monomodal Most probable: Average: Canonical EOS ≈ microcan. Bimodal: ensembles inequivalent Entropy 1 0. 1 <E> Microcanonical Canonical <E> (Most Probable) Canonical interpolates 2 stable microcan. solutions Nuclear Phases, India, 2006 10 Gas Liquid Canonical 100 Lattice-gas Model Temperature F. Gulminelli & Ph. Ch. , PRE 66 (2002) 46108 : 80 Lattice-Gas

C-Finite systems: ensemble inequivalence Many different ensembles: Constraints and boundaries Boundaries = ∞ information incompatible with max S Various ensembles are related: Laplace transform: Probabilities (sorting): But are not equivalent: Small corrections far from phase transitions Strong deviations associated with phase transitions Disappears at thermo limit, Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80

Nuclear Phases, India, 2006 : 80