phase transition in finite systems
DESCRIPTION
Phase transition in finite systems. Philippe CHOMAZ, GANIL. Finite systems Zeroes of the partition sum Bimodal event distributions Negative heat capacity Abnormal fluctuations Experimental results Melting of Na clusters Superfluidity in atomic nuclei Fragmentation of nuclei - PowerPoint PPT PresentationTRANSCRIPT
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Phase transition in finite systems
Philippe CHOMAZ, GANIL
Finite systems Zeroes of the partition sum Bimodal event distributions Negative heat capacity Abnormal fluctuations
Experimental results Melting of Na clusters Superfluidity in atomic nuclei Fragmentation of nuclei Fragmentation of H-clusters
2
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*****Chapitre 1
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- I -Introduction
Phase transitionAnomaly in thermo
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EOS:
€
<E>=−∂β logZ
Ex: first order:discontinuous
Phase transition in infinite systems
R. Balian, Springer (1982)
Order n:discontinuity in
Ehrenfest’s definition
€
∂β
nlogZ
Caloric curve
€
Z = e−βE ( n)
(n )∑ ⎛
⎝ ⎜ ⎞
⎠ ⎟
L.E. Reichl, Texas Press (1980)
€
N→ ∞Thermodynamical potentials:
€
e.g. F = −T logZ
€
βEOS
Temperature-1
Ene
rgy
<E
>
Thermodynamical potentials:non analytical at
4
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Thermodynamical potentials:non analytical at
Phase transition in infinite systems
No phase transition continuous
Ehrenfest’s definition
EOS:
€
<E>=−∂β logZ
Ex: Continuous
€
N→ ∞
€
βEOS
Temperature-1
Ene
rgy
<E
>
€
≠Caloric curve
Thermodynamical potentials: Z >0,
€
∂β
nlogZ
Phase transition in infinite systems
€
Z = e−βE ( n)
(n )∑ ⎛
⎝ ⎜ ⎞
⎠ ⎟
€
e.g. F = −T logZ
6
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But anomaly in the entropy Ex: negative heat capacity
Thermodynamical potentials:non analytical at
Phase transition in infinite systems
No phase transition continuous
Ehrenfest’s definition
€
N→ ∞
€
≠
€
βEOS
Temperature-1
Ene
rgy
<E
>
Caloric curve
Thermodynamical potentials: Z >0,
€
∂β
nlogZ
Phase transition in infinite systems
€
Z = e−βE ( n)
(n )∑ ⎛
⎝ ⎜ ⎞
⎠ ⎟
€
e.g. F = −T logZ
EnergyTem
pera
ture
E1 E2
Caloric curve
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477; Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495.; W. Thirring, Z. Phys. 235, 339 (1970), D.H.E. Gross, Rep. Prog. Phys. 53, 605 (1990),
€
C−1 = ∂ET€
T−1 = ∂E S
8
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1st order in finite systems PC & Gulminelli Phys A (2003)
Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404
Re
Im
Complex
k /
T2 Back Bending in EOS (T(E))
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495.
Energy
Tem
pera
ture
E1 E2
Caloric curve
Abnormal fluctuation (k(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
EnergyE1 E2
Ck(3/2)
€
c
€
c
€
c
€
c Free order param. (canonical)
Fixed order para. (microcanonical)
Bimodal distribution (P (E)) K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
E distribution at
Energy
10
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*****Chapitre 2
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- II -Origin of singularities
Zeroes of Z ()
Bimodal P(E)
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, then logZ remains analytical=> No phase transition
Discontinuities from zeroes of Z
Re zeroes of Z
Im
Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z
(C.N.Yang T.D.Lee 1952)
If, when , no zeroes of Z converge on the Real axis
€
N→∞
Complex plane
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, then logZ remains analytical=> No phase transition
Discontinuities from zeroes of Z
Re zeroes of Z
Im
Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z
(C.N.Yang T.D.Lee 1952)
If, when , no zeroes of Z converge on the Real axis
€
N→∞
Complex plane
0 First order phase transitions
A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Im
Re
Yang - Lee theorem
12
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0 First order phase transitions
A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Im
Re
Yang - Lee theorem
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A theoretical definition Only valid asymptotically: V = How to extend it, V ≠ ?
How to use it experimentally ?
0 First order phase transitions
A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Im
Re
Yang - Lee theorem
∞∞
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0 First order phase transitions
A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Im
Re
Yang - Lee theorem
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First order phase transitions A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Link with energy distribution P(E)
0Im
Re
€
Zβ( )=Zβ0( ) dEPβ0 E( )e−β0−β( )E∫
Z Laplace transform of P:
€
Pβ E( )=W(E)e−βE/Zβ( )
Ph. Ch., F. Gulminelli, Physica A 2003
€
Zβ( )= dEW E( )e−βE∫
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First order phase transitions A uniform density of zero's on a line crossing the real axis perpendicularly at 0
Link with energy distribution P(E)
0Im
Re
P1(E)
P2(E)
Bimodal P with EN
Z Laplace transform of P:
€
Pβ E( )=W(E)e−βE/Zβ( )
<=
> Ph. Ch., F. Gulminelli, Physica A 2003
€
Zβ( )= dEW E( )e−βE∫
€
Zβ( )=Zβ0( ) dEPβ0 E( )e+β0−β( )E∫
14
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Fin Zero=>bimodalite
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Closest neighbors interaction
Metropolis Boltzmann weight e-E
Average volume <V> (pressure)
Lattice gas model
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Canonical(Average)
Lattice-gas Model
Closest neighbors interaction
Metropolis Boltzmann weight e-E
Average volume <V> (pressure)
Lattice gas model
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
<E> smooth fonction of (canonical caloric curve)
16
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Canonical(Average)
Lattice-gas Model
Closest neighbors interaction
Metropolis Boltzmann weight e-E
Average volume <V> (pressure)
Lattice gas model
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
<E> smooth function of (canonical caloric curve)
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Canonical(Average)
Lattice-gas Model
En
erg
y D
istr
ibu
tio
n
1
10
100
0.1
Liquid Gas
Energy distribution at a fix temperature
Canonical(Most Probable)
Lattice gas model
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
18
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Canonical(Average)
Canonical(Most Probable)
Lattice-gas Model
En
erg
y D
istr
ibu
tio
n d
1
10
100
0.1
Liquid Gas
Energy distribution at a fixed temperature
Discontinuity of the most probable
Lattice-gas model
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
18
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Canonical(Average)
Canonical(Most Probable)
Lattice-gas Model
Dis
trib
uti
on
d ’
éne
rgie
1
10
100
0.1
Energy distribution at a fixed temperature
Discontinuity of the most probable
Liquid GasLattice-gas model
Bimodal distribution
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
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A first order phase transition in finite systems
A bimodal (event) distribution of energy
If the energy is free to fluctuate (canonical - energy reservoire)
<=
>
20
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A first order phase transition in finite systems
A bimodal (event) distribution of an observable (order parameter)
If this observable is free to fluctuate (intensive ensemble)
<=
>
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Access to order parameters Nature of order parameter:
Is it collective? Scaling of the jump with N:
What happen at the thermo limit (N = ∞) ?
Is it a macro. phase transition?
A bimodal (event) distribution of an observable (order parameter)
If this observable is free to fluctuate (intensive ensemble)
A1
A2
A3
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Tem
per
atur
e,
Pro
bab
ilit
y P
(E
)
Energy
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na Cluster
Liquid-gas in lattice-gas
M
Ising model
Fragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Bimodal distributions
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***** bébut Na
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Melting of Na147 .
++
energy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Variable THelium
Heat Bath
E*(T) Cluster beam
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Melting of Na147 .
++
energy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
+4h
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Melting of Na147 .
++
6h5h3h
3h +4h 6h
Number of evaporated atoms
cou
nts
+4henergy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
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Melting of Na147 .
++
3h 4h 5h 6h
Number of evaporated atoms
cou
nts
3h 4h 5h 6henergy
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
P(E
)
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Melting of Na147 .
++
3h 4h 5h 6h
Number of evaporated atoms
cou
nts
3h 4h 5h 6henergy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
Tem
per
atu
re
Number of evaporated atoms
3h 4h 5h 6h
Number of evaporated atoms
Melting
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Melting of Na147 .
++
3h 4h 5h 6h
Number of evaporated atoms
cou
nts
3h 4h 5h 6henergy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
Tem
per
atu
re
Number of evaporated atoms
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Melting of Na147 .
++
3h 4h 5h 6h
Number of evaporated atoms
cou
nts
3h 4h 5h 6henergy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
Tem
per
atu
re
Number of evaporated atoms
Energy distribution
Expected Photo-dissociation
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Melting of Na147 .
++
3h 4h 5h 6h
Number of evaporated atoms
cou
nts
3h 4h 5h 6henergy
P(E
)
M.Schmidt et al, Nature 1999, PRL 2001
Photo-disintegration
Variable THelium
Heat Bath
E*(T)
E*(T)+ nh
Laser E=nh
Cluster beam
dete
ctor
dete
ctor
Tem
per
atu
re
Number of evaporated atoms
Energy distribution
Expected Photo-dissociation
Observed
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Tem
per
atur
e,
Pro
bab
ilit
y P
(E
)
Energy
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na Cluster
Liquid-gas in lattice-gas
M
Ising model
Fragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Bimodal distributions
24
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*****Fin bimodalité
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*****Chapitre 3
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- III -Convex S
Back bending T
Negative heat capacity
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Canonical(Average)
Canonical(Most Probable)
Lattice-gas Model
En
erg
y D
istr
ibu
tio
n
1
10
100
0.1
Liquid Gas
Energy distribution at a fixed temperature
Discontinuity of the most probable
Lattice-gas model
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
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Lattice-gas model
Canonical(Average)
Canonical(Most Probable)
Lattice-gas Model
En
erg
y D
istr
ibu
tio
n
1
10
100
0.1
Liquid Gas
E distribution P =W(E)e - E/ Z
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
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Canonical(Average)
Canonical(Most Probable)
Lattice-gas Model
En
erg
y D
istr
ibu
tio
n d
1
10
100
0.1
Liquid Gas
Convex Entropy
Microcanonical
Microcanonic: S=logW log Z P =S(E) - E
Entropy
Negative Heat Capacity
Energy (A MeV)-2 -1 0 1 2 3
3.6
3.5
3.4
3.3
3.2
Tem
per
atu
re (
MeV
)
Decreasing T: T-1= dES
T-1 = - dE(logP
Lattice-gas model
26
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A first order phase transition in finite systems
A negative heat capacity
If the energy is fixed (microcanonical)
<=
>
28
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A first order phase transition in finite systems
A inverted curvature of the thermodynamical potential
If the observables (order param.) are fixed (intensive ensemble)
<=
>
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C
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Negative Heat Capacity
First Order Phase Transition in Finite Systems .
Lynden-Bell et al, -on. Not. R. Astr. 138(1968)495, cond-mat/9812172
STARS
30
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C
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Excitation Energy (MeV)
10 2 3 4 5 6
Tem
per
atu
re (
MeV
)
0
0.4
0.8
0.
0.4
0.8
0.
0.4
0.8
1.2172Yb
166Er
162Dy
Mel
by-
1999
Nuclear superfluidity
Gob
et-2
002
Fragmenting H-clusters
F. G
ulm
inel
li, P
h. C
h. P
RL
(199
8)
Canonical liquid-gas
Abnormal curvatures
Convex entropy T decreasing with E
Compressibility < 0Susceptibility < 0
30
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***** bébut superfluid
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At low energy
Melby et al, PRL 83(1999)3150
Decay: level density
X(3He, 3He ’)X*
W(Ef)
32
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C
Excitation Energy (MeV)10 2 3 4 5 6
Mic
roca
no
nic
al T
(M
eV
)
0
0.4
0.80.
0.4
0.80.
0.4
0.8
1.2
166Er
162Dy
172Yb
Breaking of pairs
Superfluid Nuclei
Melby et al, PRL 83(1999)3150
Decay: level density
Heat
Cap
aci
ty(k
B)
T(MeV)0.5 1.00.0
20.
0.
40.
172Yb
Canonical
T1 logWE
Microcanonical
Superfluid
A.Schiller et al. 2000
X(3He, 3He ’)X*
W =
exp
S
33
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C
Excitation Energy (MeV)
10 2 3 4 5 6
Tem
per
atu
re (
MeV
)
0
0.4
0.8
0.
0.4
0.8
0.
0.4
0.8
1.2172Yb
166Er
162Dy
Mel
by-
1999
Nuclear superfluidity
Gob
et-2
002
Fragmenting H-clusters
F. G
ulm
inel
li, P
h. C
h. P
RL
(199
8)
Canonical liquid-gas
Abnormal curvatures
Convex entropy T decreasing with E
Compressibility < 0Susceptibility < 0
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C
*******Fin C neg
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Negative heat capacities Abnormal energy fluctuations
Multifragmentation
- IV-
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C
Microcanonical: fluct. partial energy
Wtot = Wk Wp
E2 = 0, k
2 = p2
2 log Wtot = -1/C f(k
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuation
Canonical: fluct. EtotZtot = Zk Zp
E2 k
2 + p2
2 log Zi = Ci / i
2
2
1can
kk
C
C
−=
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C
Microcanonical:fluct. partial energy
Wtot = Wk Wp
E2 = 0, k
2 = p2
2 log Wtot = -1/C f(k
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuation
Canonical: fluct. EtotZtot = Zk Zp
E2 k
2 + p2
2 log Zi = Ci / i
2
2
1can
kk
C
C
−=
35
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C
Microcanonical:fluct. partial energy
Wtot = Wk Wp
E2 = 0, k
2 = p2
2 log Wtot = -1/C f(k
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuation
Canonical: fluct. EtotZtot = Zk Zp
E2 k
2 + p2
2 log Zi = Ci / i
2
2
1can
kk
C
C
−=
35
Excitation Energy (AMeV)
MULTICS
MULTICS
D’Agostino et al.
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C
Sort events in E* (Calorimetry)
Multifragmentation experiment
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Sort events in E* (Calorimetry)
Multifragmentation experiment
Reconstruct a freeze-out partition
• E2=
iB
i+E
coul
1- Primary fragments: mi
• <E1>=<
ia
i> T2+ 3/2 <M-1> T
3- Kinetic EOS:
• E1=E*- E
2=> <E1>, 1
T, C1 ,2 = C1 T2can
2
2
1can
kk
C
C
−= Deduce C:
Ecoul2- Freeze-out volume:
37
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Heat capacity from energy fluctuation
Large fluctuations
D’Agostino et al.
C12
C1 2/ T2 C =
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Large fluctuations
Heat capacity from energy fluctuation
Ck T2
Ck T2 2
2
D’Agostino et al.
Negative CPeripheral Au+Au eventsC1
2
C1 2/ T2 C =
39
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C
Flu
ctu
atio
n, C
1
Heat capacity from energy fluctuation
Excitation Energy (AMeV)
D’Agostino et al.
MULTICS
Hot Au, ISIS collaboration
ISIS
ISIS
MULTICS
Ni+Ni, 32 AMeV Quasi-Projectile
Excitation Energy (AMeV)
INDRA
Bougault et al Xe+Sn central
0 2 4 6 80
2
1
INDRA
2
2
1can
kk
C
C
−=Multics E1=20.3 E2=6.50.7Isis E1=2.5 E2=7.Indra E2=6.0.5
Excitation Energy (AMeV) Excitation Energy (AMeV)
Flu
ctu
atio
n, C
1H
eat
Cap
acit
y
Hea
t C
apac
ity
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Au+Au 35 EF=0
Comparison central / peripheral
Up to 35 A.MeV the flow ambiguity is a small effect
Au+Au 35 EF=1
M.D
’A
gost
inoI
WM
2001
CCkinkin==dEdEkinkin/dT/dT
peripheral
central
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***** bimodal fragments 2
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Ni+Au INDRA data O.Lopez 2001
B. Borderie, 2001
Xe+Sn INDRA data
Distribution of events.Largest fragment/second largest
Charge big - small
Bimodality in event distribution Multifragmentation of nuclei
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Bimodality in event distribution Multifragmentation of nuclei
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CO.Lopez et al. 2001, B.Tamain et al. 2003
Z1
Z2 "reservoir" "reservoir"
T
197Au
197AuAu+Au 80 A.MeVINDRA@GSI data
Z1-Z2
Z1
Order parameter Z1-Z2 (big-small)
cf
€
ρL−ρG
Largest fragment/second largest
Bimodality in event distribution Multifragmentation of nuclei
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***** Chapitre 5
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Conclusion
Negative heat capacitiesBimodality, fluctuations
- V -
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1st order in finite systems PC & Gulminelli Phys A (2003)
Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404
Re
Im
Complex
€
c Order param. free (canonical)
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1st order in finite systems PC & Gulminelli Phys A (2003)
Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404
Re
Im
Complex
k /
T2 Back Bending in EOS (T(E))
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Energy
Tem
pera
ture
E1 E2
Caloric curve
Abnormal fluctuation (k(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
EnergyE1 E2
Ck(3/2)
€
c
€
c
€
c
€
c Order par. free (canonical)
Order par. fixed (microcanonical)
Bimodal E distribution (P (E)) K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
E distribution at
Energy
36
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C
D’A
gost
ino
PL
B 2
000
Boiling nuclei
Tem
per
atur
e, P
roba
bili
ty P
(E
)
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na ClusterFragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Excitation Energy (MeV)
10 2 3 4 5 6
Tem
per
atu
re (
MeV
)
0
0.4
0.8
0.
0.4
0.8
0.
0.4
0.8
1.2172Yb
166Er
162Dy
Mel
by-
1999
Nuclear superfluidity
Gob
et-2
002
Fragmenting H-clusters
Seen in experiments
Energy
4
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C
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C
noir
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C
noir
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C
noir
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C
noir
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C
noir
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C
noir
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**** Debut V-E lattice-Gas
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Liquid-gas
Phase transitionand bimodality
Isobare ensemble
P(i) exp-V(i)
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Liquid-gas
Phase transitionand bimodality
Negative heat capacity
Isobare ensemble
P(i) exp-V(i)
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Liquid-gas
Phase transitionAnd bimodality
Negative compressibility
Negative heat capacity
Isobare ensemble
P(i) exp-V(i)
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Liquid-gas
Phase transitionAnd bimodality
Negative compressibility
Order parameter
Negative heat capacity
Isobare ensemble
P(i) exp-V(i)
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Liquid-gas
Phase transitionAnd bimodality
Negative compressibility
Order parameter
Negative heat capacity
Isobare ensemble
P(i) exp-V(i)
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C
Tem
per
atur
e,
Pro
bab
ilit
y P
(E
)
Energy
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na Cluster
Liquid-gas in lattice-gas
M
Ising model
Fragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Bimodal distributions
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C
noir
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**** Ising début
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Magnetization in Ising Model
One Phase
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Magnetization in Ising Model
One PhaseBimodalTwo Phases
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Magnetization in Ising Model
One PhaseBimodalTwo Phases
Second Order
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Magnetization in Ising Model
One PhaseBimodalTwo Phases
Second Order
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5) Ising
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Magnetization in Ising Model
Phase transitiona bifurcation
M
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C
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C
Tem
per
atur
e,
Pro
bab
ilit
y P
(E
)
Energy
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na Cluster
Liquid-gas in lattice-gas
M
Ising model
Fragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Bimodal distributions
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***** Début bimod frag
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C
Ni+Au INDRA data O.Lopez 2001
B. Borderie, 2001
Xe+Sn INDRA data
Distribution of events.Largest fragment/second largest
Charge big - small
Bimodality in event distribution Multifragmentation of nuclei
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Bimodality in event distributionMultifragmentation of nuclei
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CO.Lopez et al. 2001, B.Tamain et al. 2003
Z1
Z2 "reservoir" "reservoir"
T
197Au
197AuAu+Au 80 A.MeVINDRA@GSI data
Z1-Z2
Z1
Order parameter Z1-Z2 (big-small)
cf
€
ρL−ρG
Largest fragment/second largest
Bimodality in event distribution Multifragmentation of nuclei
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Tem
per
atur
e,
Pro
bab
ilit
y P
(E
)
Energy
Energy distribution
Photo-dissociation
3h 4h 5h
Number of evaporated atoms
Haberland-PRL 2001
Melting of Na Cluster
Liquid-gas in lattice-gas
M
Ising model
Fragment asymmetry (Zbig-Zsmall)
INDRA
Multifragmentation of nuclei
INDRA-2001
Bimodal distributions
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noir
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noir
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noir
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noir
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noir
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***** Début neg comp
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Density: Order parameter: Usually V=cst
(“box”) Isochore
ensemble Negative
compressibility
Canonical Lattice-Gas
Density 0
0 0.2 0.4 0.6 0.8
Pre
ssu
re (
MeV
/fm
3)
-4
-2
0
2
4
6
8
4
68
T=10 MeV
Canonical lattice gas at constant V
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Generalization: inverted curvatures
Negative Susceptibility and Compressibility
Canonical Lattice-Gas
Canonical Lattice-Gas
Particle Number
Pressure
Ch
emic
al P
oten
tial
Gulminelli, PC PRL99
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Excitation Energy (MeV)
10 2 3 4 5 6
Tem
per
atu
re (
MeV
)
0
0.4
0.8
0.
0.4
0.8
0.
0.4
0.8
1.2172Yb
166Er
162Dy
Mel
by-
1999
Nuclear superfluidity
Gob
et-2
002
Fragmenting H-clusters
F. G
ulm
inel
li, P
h. C
h. P
RL
(199
8)
Canonical liquid-gas
Abnormal curvatures
Convex entropy T decreasing with E
Compressibility < 0Susceptibility < 0
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noir
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***** début H clusters
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n
He
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H2
Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
He
H2H2
HH
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
He H2
H2H2
H
H
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
HeH2
H2H2
H
H
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
HeH2
H2
H2
HH
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
He
H2H2
H2
HH
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
H3+(H2)n’
He
H2H2
H2
HH
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
Partition => energy Partition => energy
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
Partition => energy
Largest => Temperature
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Fragmentation of hydrogen clusters
Gobet et al, PRL (2002)
Partition => energy
Largest => Temperature
Cooler
Hotter
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Excitation Energy (MeV)
10 2 3 4 5 6
Tem
per
atu
re (
MeV
)
0
0.4
0.8
0.
0.4
0.8
0.
0.4
0.8
1.2172Yb
166Er
162Dy
Mel
by-
1999
Nuclear superfluidity
Gob
et-2
002
Fragmenting H-clusters
F. G
ulm
inel
li, P
h. C
h. P
RL
(199
8)
Canonical liquid-gas
Abnormal curvatures
Convex entropy T decreasing with E
Compressibility < 0Susceptibility < 0
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noir
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noir
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noir
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noir
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noir
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***** Appendixe flow
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-Appendix - I -Phase transition under flows
Exact theory for radial flow Isobar ensemble required Phase transition not affected
Example of oriented flow (transparency) Exact thermodynamics Effects on partitions
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€
p(n)∝exp−βEint(n) −β
(r p i(n)−mα
r r i (n))2
2mi=1
A∑ +βmα2
2 ri(n)2
i=1
A∑
Radial flow at “equilibrium”
Equilibrium = Max S under constrains
Additional constrains: radial flow <pr(r)> Additional Lagrange multiplier (r)
Self similar expansion <pr(r)> = m r => (r) = r
Does not affect thermo is flow subtracted Gulminelli, PC, Nucl-Th (2002)
x
zExpansion
€
p(n)∝ exp−βE(n)− γ(r)r p i(n)
r r i (n)/ri(n)i=1
A∑( )
Thermal distribution in the moving frame
Negative pressure Isobar ensemble
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Radial flow at “equilibrium”
Does not affect thermo is flow subtracted Gulminelli, PC, Nucl-Th (2002)
xz
Expansion
Does no affect r partitioning
Only reduces the pressure
Shifts p distribution Changes fragment
distribution: fragmentation less effective
pres
sure
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Transparency at “equilibrium”
Equilibrium = Max S under constrains
Additional constrains: memory flow <pz> Additional Lagrange multiplier
EOS leads to <pz> = Ap0 with
Affects p space and fragments,
Does not affect the thermo if Eflow subtractedGulminelli, PC, Nucl-Th (2002)
€
p(n)∝ exp−βE(n)−α τi(n)pzi
(n)i=1
A∑
px
pzTransparency
=-1 target =1 projectile {
€
<τpz>=∂αlogZβα
€
r p 0=
r k mα/β
€
p(n)∝exp−βEint(n) −β
(r p i(n)−τi
(n)r p 0)2
2mi=1
A∑ Thermal distribution in moving frame
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Transparency at “equilibrium”
Affects p space and fragments
Gulminelli, PC, Nucl-Th (2002)
px
pzTransparency
Does no affect r partitioning
Shifts p distribution Changes fragment
distribution: fragmentation less effective
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Transparency at “equilibrium”
Affects p space and fragments
Gulminelli, PC, Nucl-Th (2002)
px
pzTransparency
Fragmentation less effective If Eflow not subtracted
Affect <E> Affect T from <Ek>
Ex: Fluctuations of Q values All thermal (from E = - 2) All relative motion
Up to 10-15% no effect
€
(T≠23<Ek>)
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noir
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***** Appendix Coulomb
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- Appendix -II-Thermo with Coulomb
Coulomb reduces coexistence and C<0 region
Statistical treatment of Coulomb (EN,VC) a common phase diagram for
charged and uncharged systems
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Coulomb reduces L-G transition
Reduces coexistence Bonche-Levit-Vautherin
Nucl. Phys. A427 (1984) 278
Reduces C<0 Lattice-Gas
OK up to heavy nuclei
A=207Z= 82
Gulminelli, PC, Comment to PRC66 (2002) 041601
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A unique framework: e-E = e-NEN -CVC
Introduce two temperatures N= and C =q2 O two energies EN and VC
Statistical treatment of Non saturating forces interactions
€
PβNβC EN,VC( )=W EN,VC( )
ZβNβCexp−βNEN−βCVC( )
C Uncharged C N Charged
€
Pβ E( )= Pβ,βC =0 E,VC( )∫ dVC
€
Pβ E( )= Pβ,β E−VC,VC( )∫ dVC
€
H =HN+q2VC- Effective charge q = 1 charged system -- Effective charge q = 0 uncharged system -
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With and without Coulomb a unique S(EN,VC)
Gulminelli, PC, Raduta, Raduta, submitted to PRL
E=EN+VC, chargedE=EN, uncharged
0.4
0.8
1.2
1.6
2C
oulo
mb
inte
ract
ion
VC
0 2 4 6 8 10 12 14
Pro
bab
ilit
y
0 4 8 12Energy
(EN ,VC) a unique phase diagram
Coulomb reduces E bimodality different weight
rotation of E axis
Energy
C N Charged
C Uncharged
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Correction of Coulomb effects
If events overlap
Gulminelli, PC, Raduta, Raduta, submitted to PRL
Reconstruction of the uncharged distribution from the charged one
Energy distribution: EntropyReconstructed
Exact
0.8
1.2
1.6
2C
oulo
mb
inte
ract
ion
VC
0 2 4 6 8 10 12 14Energy
Pro
bab
ilit
y
0 2 4 6 8 10 12 14Energy
0.8
1.2
1.6
2C
oulo
mb
inte
ract
ion
VC
C N Charged
C Uncharged
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Energy
12
34
Cou
lom
b in
tera
ctio
n V
C
0 2 4 6 8 10 12 14
C N Charged
C Uncharged
Partitions may differ for heavy nuclei Channels are
different (cf fission)
Re-weighting impossible
However, a unique phase diagram
(EN VC)
Gulminelli, PC, Raduta, Raduta, submitted to PRL
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noir
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***** Appendix equilibrium
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-Appendix - III -Equilibrium in finite systems
Macroscopic One realization (event) at equilibrium
Microscopic Statistical ensemble and information Ergodicity versus mixing dynamics Multiplicity of equilibria What is temperature?
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Equilibrium : What Does It Mean? Finite systems
In time In space
Isolated open systems No reservoir or bath No container
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Equilibrium : What Does It Mean?
Thermodynamics : infinite system
One ∞ system = ensemble of ∞ sub-systems
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Equilibrium : What Does It Mean?
Finite system
Cannot be cut in sub-systems
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if it is ergodic
Equilibrium : What Does It Mean?
One small system in time
Cannot be cut in sub-systems
Is a statistical ensemble
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if it is chaotic
if it is ergodic
Equilibrium : What Does It Mean?
One small system in time
Many events
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Dynamics: global variables Statistics: population
Many different ensembles
Statistics from minimum information
“Chaos” populates phase space
MicrocanonicalE <E> V <r3> <Q2> <p.r> <A> <L>
CanonicalIsochoreIsobareDeformedExpandingGrandRotating
... Others
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What is temperature ?T
W equiprobable microstates
€
S=klogW
L. Boltzmann
Entropy = disorder (-information)
R. Clausius
T is the entropy increase
€
dS=dQT
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What is temperature ?
The microcanonical temperatureS =T-1 ES = k logW
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What is temperature ?
It is what thermometers measure.
E = Ethermometer + Esystem
The microcanonical temperatureS =T-1 ES = k logW
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What is temperature ?
It is what thermometers measure.
E = Ethermometer + Esystem
The microcanonical temperatureS =T-1 ES = k logW
Distribution of microstates
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S = k logW
What is temperature ?
The microcanonical temperature
It is what thermometers measure.
Distribution of microstates
S =T-1 E
Eth
E
P(E
th)
0 0
E = Eth + Esys
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Most probable partition: Tth = sys
What is temperature ?
The microcanonical temperature
It is what thermometers measure.
Equiprobable microstates
max P => logWth - logWsys =0
S =T-1 E
Eth
E
P(E
th)
0 0
P(Eth) Wth (Eth)* Wsys(E-Eth)
E = Eth + Esys
S = k logW
max P => Sth - Ssys =0
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noir
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****** Appendix volume
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-Appendix -IV-C<0 in Liquid gas transition
Negative heat capacity in fluctuating volume ensemble
Difference between CP and CV
Difference between C and E(T) Role of the thermo transformation
Negative heat capacity in statistical models and Channel opening
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- a -Role of volume fluctuationsFixed volume Cv>0 but K<0 Open systems Cp<0
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Volume: order parameter L-G in a box: V=cst
Lattice-Gas Model
Negative compressibility
Density 0
0 0.2 0.4 0.6 0.8
Pre
ssu
re (
MeV
/fm
3)
-4
-
2
0
2
4
6
8
4
6
8 T=10 MeV
Canonical lattice gas at constant V
Gulminelli & PC PRL 82(1999)1402
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Volume: order parameter L-G in a box: V=cst
CV > 0
Lattice-Gas Model
Negative compressibility
Thermo limit
Density 0
0 0.2 0.4 0.6 0.8
Pre
ssu
re (
MeV
/fm
3)
-4
-
2
0
2
4
6
8
4
6
8 T=10 MeV
Canonical lattice gas at constant V
Thermo limit
Gulminelli & PC PRL 82(1999)1402
See specific discussion for large systems Pleimling and Hueller, J.Stat.Phys.104 (2001) 971 Binder, Physica A 319 (2002) 99. Gulminelli et al., cond-mat/0302177, Phys. Rev. E.
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Open systems (no box)Fluctuating volume
Bimodal P(V) Negative
compressibility
Bimodal P(E) Negative heat
capacity Cp<0
Isobar ensemble
P(i) exp-V(i)
PC, Duflot & Gulminelli PRL 85(2000)3587
Isobarcanonicallattice gas
microcanonicallattice gas
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Constrain on Vi.e. on order paramameter
Suppresses bimodal P(E) No negative heat
capacity CV > 0
See specific discussion for large systems Pleimling and Hueller, J.Stat.Phys.104 (2001) 971 Binder, Physica A 319 (2002) 99. Gulminelli et al., cond-mat/0302177, Phys. Rev. E.
Isochorecanonicallattice gas
microcanonicallattice gas
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- b -Pressure dependent EOSCaloric Curves not EOS Fluctuations measure EOS
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Volume dependent T(E)
Energy
Tem
per
atu
re
Pressu
re
2-D EOS
E TV P
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Lattice-Gas Model
Volume dependent T(E)
2-D EOS
Energy
Tem
per
atu
re
Pressu
re
E TV P
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Lattice-Gas Model
Volume dependent T(E)
2-D EOS
First orderLiquid gasPhase Transition
Critical point
Energy
Tem
per
atu
re
Pressu
re
E TV P
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Lattice-Gas Model
Volume dependent T(E)
Energy
Tem
per
atu
re
Pressu
re
Negative Heat Capacity
TransformationdependentCaloric Curve
- P = cst
- <V> = cst
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Temperature
Ck
Fluctuations
Volume dependent EOS
<V> = cst
P = cst
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Ck
P = cst <V> = cst Volume dependent EOS
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Volume dependent EOS P = cst <V> = cst
The Caloric curvedepends on thetransformation
measures EOSAbnormal inLiquid-Gas
C is not dT/dE
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Caloric curve are not EOSFluctuations are P = cst <V> = cst
The Caloric curvedepends on thetransformation
measures EOS
abnormal inLiquid-Gas
C is not dT/dE
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- c -Negative curvaturesChannel opening Ex: Statistical models
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C<0 in statistical modelsLiquid-gas and channel opening
q = 0 =>
no-Coulomb
Many channel opening
One Liquid-Gas transition (Vc related to V) 0.
40.
81.
21.
62
Cou
lom
b in
tera
ctio
n V
C
0 2 4 6 8 10 12 14Nuclear energy
Pro
bab
ilit
y
Probability
€
H =HN+q2VCBimodal P(EN)
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noir
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***** Appendix iteration et correlations
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-Appendix -V-How to progress
Correcting errors in reconstruction
Iterative procedure
C from other observables Using correlated observables
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Correction of experimental errors
2
2
1can
kk
C
C
−=
Heat capacity from fluctuations
can can be “filtered”
(apparatus and procedure)
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Correction of experimental errors
2
2
1can
kk
C
C
−=
Heat capacity from fluctuations
can can be “filtered”
(apparatus and procedure) k can be corrected
Iterate the procedure Freeze-out reconstruction Decay toward detector
Reconstructed freeze-out fluctuation
Corrected freeze-out fluctuation
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Heat capacity from any fluctuation
2
2
1can
kk
C
C
−=
From kinetic energy
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Heat capacity from any fluctuation
2
2
1can
kk
C
C
−=
From kinetic energy
From any correlated observable (Ex. Abig)
220
2 11
A
Ak
CC
−=
k k correlation “canonical” fluctuations
Ek
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noir
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***** Appendix test hypothèses
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-Appendix -VI-Test of freeze-out reconstruction
Volume Coulomb boost
Temperature Isotope ratios Particle-fragment correlations
Alternative extraction
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Multifragmentation experiment
Sort events in energy (Calorimetry)
Reconstruct a freeze-out partition
• E2=
im
i+E
coul
• E1=E* -E
2
• <E1>=<
ia
i> T2+ 3/2 <M-1> T
1- Primary fragments:2- Freeze-out volume:
mi
Ecoul
3- Kinetic EOS: T, C1
=> <E1>, 1
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Multifragmentation experiment
Sort events in energy (Calorimetry)
Reconstruct a freeze-out partition
• E2=
im
i+E
coul
• E1=E* -E
2
• <E1>=<
ia
i> T2+ 3/2 <M-1> T
1- Primary fragments:2- Freeze-out volume:
mi
Ecoul
3- Kinetic EOS: T, C1
=> <E1>, 1
E*
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C
Volume
From Coulomb boost C Weakly sensitive to V
Au QP
6V0
3V0
M.D
’A
gosti
no N
PA
2002
<E
k>
Eflow (A.MeV)
3 4 5 6V0
exp.
central Xe+Sn
R.B
ou
gau
lt 2
002
Ambiguity with expansion
Tiny influence on C
Need isotopic resolution
mass number
<E
k> Z=cte
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C
Volume
From Coulomb boost C Weakly sensitive to V
Au QP
6V0
3V0
M.D
’A
gosti
no N
PA
2002
<E
k>
Eflow (A.MeV)
3 4 5 6V0
exp.
central Xe+Sn
R.B
ou
gau
lt 2
002
Ambiguity with expansion
Tiny influence on C
Need isotopic resolution
mass number
<E
k> Z=cte
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CM.D’Agostino et al. 2001
3V0
6V0
a=12 a=8
C Ck/(Ck-k/T2)
Ek = E*-Q-Vc(V)
<Ek> = aiT2 + 3/2(<m>-1)T
Ck = d <Ek>/dT Ck<1.5 Atot
2 2Au+Au 35 A.MeV
MULTICS data
Influence on C
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C
T C isotope
M.D ’Agostino NPA2002
Evaporation and temperature
<Ek> = ai(T)T2 + 3/2(<m>-1)T
Consistency of T
Correlation
Au QPS.Hudan 2002
A/8
A/13
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Heat capacity from the translational energy
( )( )2
2 2)2/3(1)2/3(1
1
1)2/3(
tr
tr
E
mmT
C
m
ET
−−−=
−=
int* EVQEE ctr −−−=
calorimetry
Coulomb trajectories
LCP-IMF correlations
NPA699(02)795
Multics Au+Au 35A.MeV
E.M.Pearson 1985, A.Raduta 2002
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Au+Au 35 EF=0
Comparison central/peripheral
Up to 35 A.MeV the flow ambiguity is a small effect
Au+Au 35 EF=1
M.D
’A
gosti
noIW
M2
00
1
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noir