Phases

1

2
??????? 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
1
3

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

52
5
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
52
6
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
52
7
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
52
8
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
52
9
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
  

52
10
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
  

52
11

12
ThermodynamicsInformation theoryStatistical
physics
-I-

2
13

14
A-Thermo Statistical ensembles
R. Balian  Statistical mechanics 

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15
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

52
16
One macroscopic system is an ensemble

• Thermodynamics infinite system

One 8 system ensemble of 8 sub-systems
17
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
Thermo describe several realizations

• One small system in time gt statistical ensemble

20
Thermo describe several realizations

• One small system in time gt statistical ensemble

Many events
21

22
B-Observation space
R. Balian  Statistical mechanics 

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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

Geometrical interpretation of observation

• Scalar product in Observable space

• Observation

• Observation

52
27

Geometrical interpretation of observation

• Scalar product in Observable space

• Observation

HUGE (Infinite) space Classical gt N6 Quantum
gt N8
52
28

29

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

8
30

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

8
31

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
8
32

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
33

34

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

1
3
z
x
y
8
35

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

1
3
z
x
y
8
36

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

1
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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38

39
C- Time evolution
R. Balian  Statistical mechanics 

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40

R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

52
41
C- Time evolution
R. Balian  Statistical mechanics 

• State
• Classical
• Phase space Density

Quantum Density Matrix

• Dynamics
• Hamilton

Schrödinger
Liouville-von Neumann

• Liouville

52
42
C- Time evolution
R. Balian  Statistical mechanics 

• Observation
• Classical

Quantum

• Dynamics

Heisenberg (Ehrenfest)

• State

Liouville-von Neumann

• Liouville

52
43

44
C- Information and Entropy
R. Balian  Statistical mechanics 

52
45
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

46

47
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

49

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

52
50
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

52
51
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

53
52
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

53
53
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

54
54

55
DiscussionTemperature Equilibra
-II-

2
56

57
T
A- What is temperature ?

R. Clausius
58

A- What is temperature ?

• The microcanonical temperature

?S T-1 ?E
S k logW
59

A- What is temperature ?

• It is what thermometers measure.

E Ethermometer Esystem

• The microcanonical temperature

?S T-1 ?E
S k logW
60

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
61

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
62

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
63

Realisation 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)
logWsys (E-Eth) logWsys (E)- Eth/T
64

65
B- What are the various equilibria?

66
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

67
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
68
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 )

69
Validity conditions
R. Balian  Statistical mechanics 

70
Information theory for finite system
R. Balian  Statistical mechanics 

71
Many different ensembles
Microcanonical
E
ltEgt
Canonical
V
Isochore
ltr3gt
Isobare
ltQ2gt
Deformed
ltp.rgt
Expanding
ltAgt
Grand

ltLgt
Rotating
...
Others
72
C-Finite systems ensemble inequivalence

73
C-Finite systems ensemble inequivalence
R. Balian  Statistical mechanics 

• Géneral ref.

• Phase trans.

74
C-Finite systems ensemble inequivalence
R. Balian  Statistical mechanics 

75
Inequivalence

76
C-Finite systems ensemble inequivalence

77

78

79

80

81

82

83
A-Statistical ensembles
R. Balian  Statistical mechanics 

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

(Variational principle)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

52
84
B-Thermodynamics
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

52
85
B-Thermodynamics
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

53
86
B-Thermodynamics
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

53
87
B-Thermodynamics 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

54
88

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

55
89

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

• Mean-field approximation
• Fermi
• Skyrme force (SLy4)
• density, kinetic density, energy
  density
• mean field,
  effective mass

55
90

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

• Mean-field approximation
• Fermi
• Skyrme force (SLy4)
• density, kinetic density, energy
  density
• mean field,
  effective mass

55
91

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

Fold
Liquid
Gas
mp
mn
56
92

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
56
93

C-Phase transition
Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
56
94

C-Phase transitions

57
95

C-Phase transitions

58
96

C-Phase transitions

Température (Degrés)
Chaleur (Calories par grammes)
58
97

C-Phase transitions

58
98

C-Phase transition
Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
58
99
Chapitre 2

58
100
D-Isospin in coexistence

59
101
D-Isospin in coexistence
Neutron density

Liquid
rn
Gas
mn
mp
Proton density
Liquid
rp
Gas
mp
mn
59
102
D-Isospin in coexistence distillation
Neutron density

• Z/A order parameter

(Except symmetric matter)
Liquid
rn
Gas
mn
Liquid
mp
lt
Proton density
Gas
Liquid
rp
Gas
mp
mn
59
103
D-Isospin in coexistence distillation

• Z/A order parameter

(Except symmetric matter)
Liquid
Gas
59
104
D-Isospin in coexistence distillation

• Z/A order parameter

(Except symmetric matter)
Liquid
Liquid symmetric
Gas
Gas asymmetric

• Isospin distillation

59
105
D-Isospin in coexistence distillation

• Z/A order parameter

Sc Scbmtsr
rp (fm-3)
(Except symmetric matter)
Liquid
Liquid
Gas
Gas
rn (fm-3)
Liquid symmetric

• Isospin distillation

Gas asymmetric
59
106

E-Role of the force

• Coexistence

T10 MeV
rp (fm-3)
rn (fm-3)
60
107
E-Role of the force

• Coexistence

SLy230a
T10 MeV
T10 MeV
rp (fm-3)
rn (fm-3)
60
108
E-Role of the force

• Coexistence

Strong force dependence
SLy230a
SGII
SIII
T10 MeV
60
109
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
110
E-Role of the force

• Coexistence

Strong force dependence
60
111
E- Role of the force

• Coexistence

Strong force dependence
T10 MeV
X
X
X
X
X
X
X
X
60
112
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
113
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
114
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
115
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
116
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
117
E-Role of the force

• Coexistence

T10 MeV
Strong force dependence
T10 MeV
T10 MeV
SLy230a
SGII
SIII
60
118

119
Interactions between nucleonsSymmetriesGeneral
properties
-II-

7
120

121

C) Radial (and momentum) dependences

• Ex Argonne v18
• Like a
• Van der Waals
• Long range
• attractif
• Short range
• hard core

Wiringa95
J. Dobaczewski, nucl-th/0301069
14
122

123

124

125

126
Ground state propertiesTheoretical
toolsMany-body problem
-III-

16
127

128

129
Independent particle motionMean fieldDensity
functional theory
-IV-

21
130

131

132

133

134

29
135
??????? Theory

• Complex systems
• NN Interaction
• Symmetries
• Solution of N-Body
• Methodes, variational
• Experimental guides
• Independent particles
• Effective interactions
• Density functional
• Nuclear matter
• Isospin
• T Phase transition
• Compact stars
• Beyond mean-field
• Symmetry breaking
• Collective coordinate
• Time dependence

Philippe CHOMAZ - GANIL
1
136
??????? Theory

• Complex systems
• NN Interaction
• Symmetries
• Solution of N-Body
• Methodes, variational
• Experimental guides
• Independent particles
• Effective interactions
• Density functional
• Nuclear matter
• Isospin
• T Phase transition
• Compact stars
• Beyond mean-field
• Symmetry breaking
• Collective coordinate
• Time dependence

Philippe CHOMAZ - GANIL
1
137

138

B) Mean field Slater trial state

• Independent particles
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number

30
139

B) Mean field Slater trial state

• Independent particles
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

30
140

B) Mean field Slater trial state

• Independent particles
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

• Fock space
• Basis Slaters
• creation operator

30
141

B) Mean field Slater trial state

• Independent particles
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

• Fock space
• Basis Slaters
• creation operator
• Basis of operators

30
142

B) Mean field Slater trial state

• Independent particles
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

• Fock space
• Basis Slaters
• creation operator
• Basis of operators

30
143

B) Mean field One body density

• Independent particles, Slater trial state
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

31
144

B) Mean field One body density

• Independent particles, Slater trial state
• Each particle in an orbital
• Fermions gt antisymmetrized
• Occupation number
• Second quantization

• One body density
• Projector on occupied states
• ltgt Slater
• Result of an observation
• ltgt 1-body operators

31
145

B) Mean field

32
146

B) Mean field Variational principal

• Minimum of ltHgt
• Variation of each orbital
• gt variation of
• Energy (2-body V)
• gt variation of E
• gt mean field
• Min with constraints
• gtHartree-Fock

32
147

B) Mean field Variational principal

• Minimum of ltHgt
• Variation of each orbital
• gt variation of
• Energy (2-body V)
• gt variation of E
• gt mean field
• Min with constraints
• gtHartree-Fock

32
148

B) Mean field Variational principal

• Minimum of ltHgt
• Variation of each orbital
• gt variation of
• Energy (2-body V)
• gt variation of E
• gt mean field
• Min with constraints
• gtHartree-Fock

32
149

B) Mean field Variational principal

32
150

B) Mean field theory

• Ex 2-body force
• Hartree-Fock Eq
• Hartree
• gtlocal
• Fock (exchange)
• gtnon-local
• Divergences if hard core VNN gt Veff

33
151

B) Mean field effective interactions

• Independent particles not valid at short distance

34
152

B) Mean field effective interactions

• Independent particles not valid at short distance
• From V to G

(2-body problem)

• Solution
• gtBethe-Goldstone
• Brueckner

(in medium a,b unoccupied)
34
153

B) Mean field

34
154

C) Mean field Density functional theory

• The only information needed is the energy
• gt functionals of r
• Local density approximation
• Energy density functional
• Local densities
• matter , kinetic , current
• Mean field

35
155

C) Mean field Density functional theory

• The only information needed is the energy
• gt functionals of r
• Local density approximation
• Energy density functional
• Local densities
• matter , kinetic , current
• Mean field

35
156

C) Mean field

32
157

C) Mean field Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional

36
158

C) Mean field Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional
• Mean-field q(n,p)

36
159

C) Mean field Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional
• Mean-field q(n,p)

36
160

C) Mean field Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional
• Mean-field q(n,p)

36
161

C) Mean field

36
162

D) Mean field Nuclear Matter

• Uniform infinite spin saturated matter
• gt Fermi Sphere

(h3 volume in phase space)
37
163

D) Mean field
SLy230a

• Equation of state

• Energy per particle
• Kinetic
• Symmetric matter
• Asymmetry

,
38
164

D) Mean field
SLy230a

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

,
39
165

D) Mean field
SLy230a

• Equation of state
• Saturation
• Compressibility
• Isopin dependence
• Effective mass

,
39
166

D) Mean field constraints on forces

• Equation of state
• Saturation
• Compressibility
• Isopin dependence
• Effective mass

40
167

168
Nuclear Matter propertiesIsospin dependence
Temperature dependence
-V-

41
169

41
170

A-Nuclear matter

42
171

A-Nuclear matter
SLy230a

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

42
172

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

42
173

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

43
174

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Liquid-drop binding -B/AEsEsurfEcoulEsym

44
175

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Liquid-drop binding -B/AEsEsurfEcoulEsym Poor
ly known close to drip-lines (low density)

44
176

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Poorly known r?rs

45
177

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Liquid-drop shape Isospin dependence of
saturation poorly known
Poorly known r?rs

45
178

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Poorly known r?rs

46
179

A-Nuclear matter
Monopole vibration

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Uncertainty Ksym gt K225-270
Poorly known r?rs

46
180

A-Nuclear matter
Monopole vibration

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Uncertainty Ksym gt K225-270
Poorly known r?rs

46
181

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Example Skyrme
Poorly known r?rs

47
182

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Density dependence of Esym Poorly known
Poorly known r?rs

48
183

A-Nuclear matter

• Equation of state
• Saturation
• Compressibility
• Isopin dependence

Density dependence of Esym Poorly known
Poorly known r?rs

49
184

49
185

186

187

188

29
189
??????? Theory

• Complex systems
• NN Interaction
• Symmetries
• Solution of N-Body
• Methodes, variational
• Experimental guides
• Independent particles
• Effective interactions
• Density functional
• Nuclear matter
• Isospin
• T Phase transition
• Compact stars
• Beyond mean-field
• Symmetry breaking
• Collective coordinate
• Time dependence

Philippe CHOMAZ - GANIL
1
190
??????? Theory

• Complex systems
• NN Interaction
• Symmetries
• Solution of N-Body
• Methodes, variational
• Experimental guides
• Independent particles
• Effective interactions
• Density functional
• Nuclear matter
• Isospin
• T Phase transition
• Compact stars
• Beyond mean-field
• Symmetry breaking
• Collective coordinate
• Time dependence

Philippe CHOMAZ - GANIL
1
191

29
192

B-Phase diagram

193

B-Phase diagram

50
194

B- Phase diagram

• Equation of state
• Saturation
• Compressibility
• Isopin dependence
• Density dependence
• Esym poorly known

50
195

B- Phase diagram

50
196

B-Phase diagram

• Condensed Fermi fluid
• Liquid-gas phase transition

• Saturation

Phase diagram
50
197
Dense matter EOS
Plasma of Quarks and Gluons
20 200 MeV
Critical points (second order)
Temperature
Nucleus
Density r/r0
1 5?
51
198
Dense matter EOS
Crab nebula
July 5, 1054
51
199
Dense matter EOS
Crab nebula
July 5, 1054
51
200
Dense matter EOS
Plasma of Quarks and Gluons
20 200 MeV
Crab nebula
July 5, 1054
Temperature
Density r/r0
1 5?
Nucleus
51
201

Les chemins de la Nucléosynthèse

Super-nova
202

Dense exotic matter in the cosmos Supernovae and
Neutron stars
Les chemins de la Nucléosynthèse

Super-nova
203


Les chemins de la Nucléosynthèse

Super-nova
204


Les chemins de la Nucléosynthèse

Super-nova
205
Dense matter EOS
Plasma of Quarks and Gluons
20 200 MeV
Crab nebula
July 5, 1054
Temperature
Density r/r0
1 5?
Nucleus
51
206
B-Thermodynamics
R. Balian  Statistical mechanics 

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

(Variational principle)

• Lagrange multipliers

Boltzman distribution
Partition sum
Equation of state

52
207
B-Thermodynamics
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

52
208
B-Thermodynamics
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

53
209
B-Thermodynamics
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

53
210
B-Thermodynamics 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

54
211

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

55
212

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

• Mean-field approximation
• Fermi
• Skyrme force (SLy4)
• density, kinetic density, energy
  density
• mean field,
  effective mass

55
213

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

• Mean-field approximation
• Fermi
• Skyrme force (SLy4)
• density, kinetic density, energy
  density
• mean field,
  effective mass

55
214

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

Fold
Liquid
Gas
mp
mn
56
215

B-Thermodynamics mean field

• Grand potential
• 2 fluids (protons and neutrons)

Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
56
216

C-Phase transition
Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
56
217

C-Phase transitions

57
218

C-Phase transitions

58
219

C-Phase transitions

Température (Degrés)
Chaleur (Calories par grammes)
58
220

C-Phase transitions

58
221

C-Phase transition
Fold
Liquid
Gas
mp
mn
Liquid
jump

• Discontinuity in r
• First order
• Liquid-gas

rp
Gas
mn
mp
58
222
Chapitre 2

58
223
D-Isospin in coexistence

59
224
D-Isospin in coexistence
Neutron density

Liquid
rn
Gas
mn
mp
Proton density
Liquid
rp
Gas
mp
mn
59
225
D-Isospin in coexistence distillation
Neutron density

• Z/A order parameter

(Except symmetric matter)
Liquid
rn
Gas
mn
Liquid
mp
lt
Proton density
Gas
Liquid
rp
Gas
mp
mn
59
226
D-Isospin in coexistence distillation

• Z/A order parameter

(Except symmetric matter)
Liquid
Gas
59
227
D-Isospin in coexistence distillation

• Z/A order parameter

(Except symmetric matter)
Liquid
Liquid symmetric
Gas
Gas asymmetric

• Isospin distillation

59
228
D-Isospin in coexistence distillation

• Z/A order parameter

Sc Scbmtsr
rp (fm-3)
(Except symmetric matter)
Liquid
Liquid
Gas
Gas
rn (fm-3)
Liquid symmetric

• Isospin distillation

Gas asymmetric
59
229

E-Role of the force

• Coexistence

T10 MeV
rp (fm-3)
rn (fm-3)
60
230
E-Role of the force

• Coexistence

SLy230a
T10 MeV
T10 MeV
rp (fm-3)
rn (fm-3)
60
231
E-Role of the force

• Coexistence

Strong force dependence
SLy230a
SGII
SIII
T10 MeV
60
232
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
233
E-Role of the force

• Coexistence

Strong force dependence
60
234
E- Role of the force

• Coexistence

Strong force dependence
T10 MeV
X
X
X
X
X
X
X
X
60
235
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
236
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
237
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
238
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
239
E-Role of the force

• Coexistence

Strong force dependence
T10 MeV
60
240
E-Role of the force

• Coexistence

T10 MeV