Phases

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??????? Phases

• 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

Philippe CHOMAZ - GANIL
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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

• First principle energy conservation
• Time independent laws (symmetry) gt E conserved

• Classical
• Point in phase space

• Quantum
• Vector of Hilbert space

?(q)??q??
q
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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

• First principle energy conservation
• Time independent laws (symmetry) gt E conserved

• Classical
• Point in phase space

• Quantum
• Vector of Hilbert space

?(q)??q??
t0
q
t0
t0
t0
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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

• First principle energy conservation
• Time independent laws (symmetry) gt E conserved

• Classical
• Point in phase space

• Quantum
• Vector of Hilbert space

?(q)??q??
t0
q
t0
t0
t0
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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

• Initial condition gt infinite information needed
• Infinite accuracy needed (Chaos)

• Classical
• 6.N coordinates
• Point in phase space

• Quantum
• 2.8 coordinates
• Vector of Hilbert space

?(q)??q??
q
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Absolute necessity of the second principle
R. Balian  Statistical mechanics 

• Initial condition gt infinite information needed
• Infinite accuracy needed (Chaos)

• Classical
• 6.N coordinates
• Point in phase space

• Quantum
• 2.8 coordinates
• Vector of Hilbert space

• Degree of freedom gt infinite information needed
• Our ignorance of initial comdition should be
  taken into account to make the theory meaningful
  

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Classical Chaos lt Quantum 8 D. freedom

• Classical
• 6.N coordinates
• Chaos

• Quantum
• 2.8 coordinates
• Projection

ltpgt
ltqgt
ltp2gt
ltqpgt
ltq2gt

• Our ignorance of initial comdition should be
  taken into account to make the theory meaningful
  

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ThermodynamicsInformation theoryStatistical
physics
-I-

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A-Thermo Statistical ensembles
R. Balian  Statistical mechanics 

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A-Ensembles
R. Balian  Statistical mechanics 

• Ensemble of events / partitions / replicas

• State
• Classical
• Point in phase space
• Ensemble statesoccurrence probability
  
• gt Phase space density

• Quantum
• Vector of Hilbert space

gt Density Matrix

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One macroscopic system is an ensemble

• Thermodynamics infinite system

One 8 system ensemble of 8 sub-systems
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A single microscopic system ? ensemble

• Finite system

Cannot be cut in sub-systems
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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

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Thermo describe several realizations

• One small system in time gt statistical ensemble

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Thermo describe several realizations

• One small system in time gt statistical ensemble

Many events
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B-Observation space
R. Balian  Statistical mechanics 

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R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

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B-Observation space
R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

• Observables
• Phase space functions

Operators (Matrices)

• Observation

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• Observation

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Geometrical interpretation of observation

• Scalar product in Observable space

• Observation

• Observation

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Geometrical interpretation of observation

• Scalar product in Observable space

• Observation

HUGE (Infinite) space Classical gt N6 Quantum
gt N8
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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) j(r) ltrjgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm

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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) j(r) ltrjgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm

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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) j(r) ltrjgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm
• gt Infinite basis

Infinite space
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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
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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) j(r) ltrjgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm
• Spin
• Hilbert basis gt gt,-gt c() ltcgt cgt
  
• Operators gt O o o .s, s (s ,s ,s ) Pauli
  matrices

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z
x
y
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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) f(r) ltrfgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm
• Spin
• Hilbert basis gt gt,-gt c() ltcgt cgt
  
• Operators gt O o o .s, s (s ,s ,s ) Pauli
  matrices

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3
z
x
y
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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) f(r) ltrfgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm
• Spin
• Hilbert basis gt gt,-gt c() ltcgt cgt
  
• Operators gt O o o .s, s (s ,s ,s ) Pauli
  matrices
• Isospin
• Hilbert basis gt ngt,pgt z( ) lt zgt
  zgt

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3
z
x
y
n
n
p
p
Physics _at_ GANIL, 2005
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Basis of operator space (1 particle)

• Spatial
• Hilbert basis gt rgt (or pgt) f(r) ltrfgt
• Operators gt ltrOrgt (or ltpOpgt)
• gt O f(r,p) ?oijklmn xiyjzkplpnpm
• Spin
• Hilbert basis gt gt,-gt c() ltcgt cgt
  
• Operators gt O o o .s, s (s ,s ,s ) Pauli
  matrices
• Isospin
• Hilbert basis gt gt,-gt z() ltzgt zgt
  
• Operators gt O o o .t, t (t ,t ,t ) Pauli
  matrices

s
s
s






1
3
z
x
y
t
t
t






1
3
Z
X
Y
Physics _at_ GANIL, 2005
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C- Time evolution
R. Balian  Statistical mechanics 

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R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

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C- Time evolution
R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

• Dynamics
• Hamilton

Schrödinger
Liouville-von Neumann

• Liouville

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C- Time evolution
R. Balian  Statistical mechanics 

• Observation
• Classical

Quantum

• Dynamics

Heisenberg (Ehrenfest)

• State

Liouville-von Neumann

• Liouville

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C- Information and Entropy
R. Balian  Statistical mechanics 

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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

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D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 

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D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 

• Gibbs equilibria are minimum bias distributions
• gt distribution maximizing the entropy
• Example
• Nothing known gt States equiprobable
• gt Microcanonical

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D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 
R. Balian  Statistical mechanics 

• Statistical ensemble
• Equilibrium Max S
• Constraints (ltHgt, ltNgt)

(Variational principle)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

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Equilibrium ensembles
D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 
R. Balian  Statistical mechanics 

• Statistical ensemble
• Equilibrium Max S
• Constraints (ltHgt, ltNgt)

(Minimum information)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

• Equilibrium entropy

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B-Thermodynamics
D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 
R. Balian  Statistical mechanics 

• Statistical ensemble
• Equilibrium Max S
• Constraints (ltHgt, ltNgt)

(Variational principle)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

• Equilibrium entropy

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B-Thermodynamics
D- Equilibrium et minimum bias (max S)
R. Balian  Statistical mechanics 
R. Balian  Statistical mechanics 

• Statistical ensemble
• Equilibrium Max S
• Constraints (ltHgt, ltNgt)

(Variational principle)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

• Equilibrium Z

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Example mean field
R. Balian  Statistical mechanics 

• Trial state
• Sc functional of r
• Max constrained S
• gtlike in a mean field
• gtEquilibrium OW
• gtFermi-dirac statistic
• gtMean field entropy

(Independent particles)

(Variational principle)

• Best approximation Sc
• gtBest approximation logZ

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DiscussionTemperature Equilibra
-II-

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T
A- What is temperature ?

R. Clausius
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A- What is temperature ?

• The microcanonical temperature

?S T-1 ?E
S k logW
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A- What is temperature ?

• It is what thermometers measure.

E Ethermometer Esystem

• The microcanonical temperature

?S T-1 ?E
S k logW
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A- What is temperature ?

• It is what thermometers measure.

E Ethermometer Esystem
Distribution of microstates

• The microcanonical temperature

?S T-1 ?E
S k logW
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A- What is temperature ?

• It is what thermometers measure.

E Eth Esys
Distribution of microstates

• The microcanonical temperature

S k logW
?S T-1 ?E
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A- What is temperature ?

• It is what thermometers measure.

E Eth Esys
Equiprobable microstates
P(Eth) ? Wth (Eth) Wsys(E-Eth)
max P gt ?logWth - ?logWsys 0
max P gt ?Sth - ?Ssys 0
Most probable partition Tth ?sys

• The microcanonical temperature

?S T-1 ?E
S k logW
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Realization of a cononical ensemble

• The thermometers is canonically distributed

E Eth Esys
Equiprobable microstates
P(Eth) ? Wth (Eth) Wsys(E-Eth)
Small thermometer (Eth small)
Ssys (E-Eth) Ssys (E)- Eth/T
Ssys logWsys
Boltzmann
P(Eth) ? Wth (Eth) exp(-Eth/T)
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B- What are the various equilibria?

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B- What are the various equilibria?
R. Balian  Statistical mechanics 

• Macroscopic
• One realization (event) can be an equilibrium
• One 8 system 8 ensemble of 8 sub-systems

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B- What are the various equilibria?
R. Balian  Statistical mechanics 

• Macroscopic
• One realization (event) can be an equilibrium
• One 8 system 8 ensemble of 8 sub-systems

• Microscopic
• Ensemble of replicas needed
• One realization (event) cannot be an equilibrium
• Gibbs Equilibrium maximum entropy
• Average over time if ergodic
• Average over events if chaotic/stochastic
• Average over replicas if minimum info

Ergodic some times used instead of uniform
population of phase space
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B- What are the various equilibria?
R. Balian  Statistical mechanics 

• Ergodic (Bound systems only)
• 8 time average phase space average
• Ergodic gt lt? statistics
• Only conserved quantities (E, J, P )

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Validity conditions
R. Balian  Statistical mechanics 

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Information theory for finite system
R. Balian  Statistical mechanics 

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Many different ensembles
Microcanonical
E
ltEgt
Canonical
V
Isochore
ltr3gt
Isobare
ltQ2gt
Deformed
ltp.rgt
Expanding
ltAgt
Grand

ltLgt
Rotating
...
Others
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C-Finite systems ensemble inequivalence

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C-Finite systems ensemble inequivalence
R. Balian  Statistical mechanics 

• Géneral ref.

• Phase trans.

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C-Finite systems ensemble inequivalence
R. Balian  Statistical mechanics 

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Inequivalence

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C-Finite systems ensemble inequivalence

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