phase transitions in nuclear reactions
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
Phase Transitions in Nuclear Reactions
Definition of phase transitions Non analytical thermo potential Negative heat capacity (1st order)
Liquid gas case C<0 and Volume (order parameter) fluctuations
Thermo of nuclear reactions Coulomb reduces coexistence and C<0 region Statistical description of radial flow
Conclusion and application to data
Ph. Chomaz and F. Gulminelli, Caen France
€
N→ ∞
€
N≠∞
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
I
-I-Definition of phase transition
Discontinuity in derivatives of thermo potential when
1st order equivalent to negative heat capacities when
€
N→ ∞
€
N≠∞
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Phase transition in infinite systems
Thermodynamical potentialsnon analytical
L.E. Reichl, Texas Press (1980)
€
F =T logZ( )
€
N→ ∞( )
Order of transition:discontinuity
Ehrenfest’s definition
Ex: first order:EOS discontinuous
R. Balian, Springer (1982)
€
<E>=−∂β logZ( )
€
(∂β
nlogZ)
Energy
Tem
pera
ture
E1 E2
Caloric curve
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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 param. free (canonical)
Order param. fixe (microcanonical)
Bimodal E distribution (P (E)) K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
E distribution at
Energy
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
II
-II-Liquid gas transition
Negative heat capacity in fluctuating volume ensemble
Difference between CP and CV
Negative heat capacity in statistical models
Neg. C and Channel opening
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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
HIC03, Montreal 2003 1 C32
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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C<0 in statistical modelsLiquid-gas and channel opening
q = 0 =>
no-Coulomb
Many channel opening
One Liquid-Gas transition
0.4
0.8
1.2
1.6
2C
oulo
mb
inte
ract
ion
VC
0 2 4 6 8 10 12 14Nuclear energy
Pro
bab
ilit
y
Probability
€
H =HN+q2VCBimodal P(EN)
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HIC03, Montreal 2003 1 C32
III
-III-Thermo of nuclear reaction
Coulomb reduces coexistence and C<0 region
Statistical treatment of Coulomb (EN,VC) a common phase diagram for
charged and uncharged systems
Statistical treatment of radial flow Isobar ensemble required Phase transition not affected
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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( )
CUncharged CNCharged
€
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
CNCharged
CUncharged
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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
CNCharged
CUncharged
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Energy
12
34
Cou
lom
b in
tera
ctio
n V
C
0 2 4 6 8 10 12 14
CNCharged
CUncharged
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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HIC03, Montreal 2003 1 C32
Equilibrium = Max S under constrains
Additional constrains: radial flow <pr(r)> Additional Lagrange multiplier (r)
Self similar expansion <pr(r)> = m r => (r) = r
Requires a confining constrain => isobar ensemble
Thermal distribution in the moving frame
Flow inducesnegative pressure
€
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”
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∑
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Radial flow at “equilibrium”
Requires a confining constrain => isobar ensemble Gulminelli, PC, Nucl-Th (2002)
x
zExpansion
Does no affect r partitioning
Only reduces the pressure
Shifts p distribution Changes fragment
distribution: fragmentation less effective
pres
sure
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
IV
-IV-Conclusion and application to data
Exact theory of phase transition Negative heat capacity in finite systems
Liquid-gas and role of volume Negative heat capacity in open systems
Role of Coulomb C<0 phenomenology qualitatively
preserved
Role of flow Thermodynamics of the isobar ensemble
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Microcanonical: partial energy
Wtot = Wk Wp
E2 = 0, k
2 =p2
2log Wtot = -1/Cf(k
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuation
Canonical: total energyZtot = Zk Zp
E2k
2 +p2
2log Zi = Ci /i
2
2
1can
kk
C
C
−=
HIC03, Montreal 2003 1 C32
Microcanonical: partial energy
Wtot = Wk Wp
E2 = 0, k
2 =p2
2log Wtot = -1/Cf(k
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuation
Canonical: total energyZtot = Zk Zp
E2k
2 +p2
2log Zi = Ci /i
2
2
1can
kk
C
C
−=
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A robust signal 2/T2T
V=cte
p=cte
Out of equilibrium
Depends only on the state
Tsalis ensemble
With flow (20%)px
pzTransparency
x
zExpansion
PC, Duflot & Gulminelli PRL 85(2000)3587
Gulminelli, PC Phisica A (2002) Gulminelli, PC Nucl-th 2002
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Sort events in energy (Calorimetry)
Multifragmentation experiment
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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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
0 2 4 6 8Excitation Energy (AMeV)
Flu
ctu
atio
n, H
eat
Cap
acit
y
0
2
1
INDRA
INDRA
Bougault et al Xe+Sn central
2
2
1can
kk
C
C
−=Multics E1=20.3 E2=6.50.7Isis E1=2.5 E2=7.Indra E2=6.0.5
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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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Heat capacity from any fluctuation
2
2
1can
kk
C
C
−=
From kinetic energy
HIC03, Montreal 2003 1 C32
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
−=
kkcorrelation “canonical” fluctuations
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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)
HIC03, Montreal 2003 1 C32
Correction of experimental errors
2
2
1can
kk
C
C
−=
Heat capacity from fluctuations
can can be “filtered”
(apparatus and procedure) kcan be corrected
Iterate the procedure Freeze-out reconstruction Decay toward detector
Reconstructed freeze-out fluctuation
Corrected freeze-out fluctuation
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Heat Bath = tr
s
No Heat Bath E Etr
Order parameters: ,
€
Aasy= AA>Acut
∑ − AA≤Acut
∑
€
Abig
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
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- 1b -Equivalence betweenbimodality of P(b) Zero's of the partition sum Z(b)
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A alternative definition in the "intensive" ensemble
First order phase transition in finite systems
is defined by
A uniform density of zero's of the
partition sum on a line crossing the real axis
perpendicularly
Yang Lee unit circle theorem
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Zero’s of Z
Partition sum
Monomodal distribution
Saddle point approximation
P1(E)
€
Zβ( )=Zβ0( ) dbPβ0 E( )e∫−β−β0( )E
€
Pβ0 E( )
€
Zβ( )=e2φ β( )= 2πσ1a1e−β−β0( )E1−1
2 β−β0( )2σ1
2
K.C.Lee 1996, 2000
P2(b)
€
Pβ E( )=eS(E)−βE
Zβ( )
E1
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Zero’s of Z
Partition sum
Bimodal distribution
Saddle point approximation
€
Zβ( )=Zβ0( ) dbPβ0 E( )e∫−β0−β( )E
€
i(2k+1)π2=12loga1
a2−β−β0( )
E20−E1
0
2 −β−β0( )2σ2
2−σ12
4K.C.Lee 1996, 2000
Re
Im
P2(E)
Re(exp(t))
Im(exp(t))
P1(E)
€
Pβ0 E( )
€
Pβ E( )=eS(E)−βE
Zβ( )
E1 E2
+
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Zero’s of ZFactorize the N zero’s [-
Inverse Laplace
€
βn=β0+i 2n+1( ) πNδ
€
Pβ+ε E( )=eεNδ /2gβ0+ε E+Nδ/2( )+e−εNδ /2gβ0+ε E−Nδ/2( )
Ph. Ch., F. Gulminelli, 2003
Re
Im
€
Z(β)=2cosh(β−β0)Nδ2( ) f(β)
€
Pβ0+ε E( )=12π dηZβ0+ε+iη( )e∫
(ε+iη)E
2 distributions splitted by the latent heat N
€
gβ0+ε E( )=12π dηf β0+ε+iη( )e∫
(ε +iη)E
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
IV
-3c-Phase transition under flows
Example of oriented flow (transparency) Exact thermodynamics Effects on partitions
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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
HIC03, Montreal 2003 1 C32
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
HIC03, Montreal 2003 1 C32
Transparency at “equilibrium”
Affects p space and fragments
Gulminelli, PC, Nucl-Th (2002)
px
pzTransparency
Fragmentation less effective Affect observables if Eflow not
subtracted Link between <Epot> and <E>
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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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
IV
-V-Phase transition other signal
Bimodalities and conservation laws Diversity of order parameters in finite
systems
Re-interpretation of fluctuation Possible experimental corrections New measures of heat capacities
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No Heat Bath E Etr
Heat Bath = tr
s
Order parameters: ,
€
Aasy= AA>Acut
∑ − AA≤Acut
∑
€
Abig
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
- 2b -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
HIC03, Montreal 2003 1 C32
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
HIC03, Montreal 2003 1 C32
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
HIC03, Montreal 2003 1 C32
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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HIC03, Montreal 2003 1 C32
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)
HIC03, Montreal 2003 1 C32
Liquid-gas
Phase transitionAnd bimodality
Negative compressibility
Order parameter
Negative heat capacity
Isobare ensemble
P(i) exp-V(i)
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Multifragmentation
Statistical Model(SMM-Raduta)
Bimodalenergydistribution
Channelopenings
Isobare ensemble
P(i) exp-V(i)
= 3.7 MeVA = 50Z = 23
Liquid Gas
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
Equilibrium : What Does It Mean?
Thermodynamics : one ∞ 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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Statistics from minimum information
“Chaos” populates phase space
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Statistics from minimum information
“Chaos” populates phase space
Compatible with freeze-out
Correct forevaporation
Subtractpre-equili-brium
BNV
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Statistics from minimum information
“Chaos” populates phase space
HIC03, Montreal 2003 1 C32
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>
CanonicalIsochoreIsobareDeformed
ExpandingExpandingGrand
RotatingRotating
... Others
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HIC03, Montreal 2003 1 C32
1. Temperature
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What is temperature ?T
W equiprobable microstates
€
S=klogW
L. Boltzmann
Entropy: missing 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 + Ebath
The microcanonical temperatureS =T-1 ES = k logW
HIC03, Montreal 2003 1 C32
What is temperature ?
It is what thermometers measure.
E = Ethermometer + Ebath
The microcanonical temperatureS =T-1 ES = k logW
Distribution of microstates
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Most probable partition: Tthermo = bath
What is temperature ?
The microcanonical temperature
It is what thermometers measure.
Distribution of microstates
max P => logWthermo - logWbath =0
S =T-1 E
Ethermo
E
P(E
ther
mo)
0 0
P(Ethermo) Wthermo * Wbath
E = Ethermo + Ebath
S = k logW
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
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HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
Heat (Calories per grams)T
emp
erat
ure
(Deg
rees
)
Phase transition in infinite systems
Thermodynamical potentialsnon analytical
Reichl or Diu or …
€
F =T logZ( )
€
N→ ∞( )
Order of transition:discontinuity
Ehrenfest’s definition
Ex: first order:EOS discontinuous
R. Balian
€
<E>=−∂β logZ( )
€
∂β
nlogZ( )
HIC03, Montreal 2003 1 C32
Phase transition in infinite systems
Thermodynamical potentialsnon analytical
Reichl or Diu or …
€
F =T logZ
€
N→ ∞
Order of transition =>discontinuous derivative
Ehrenfest’s definition
Ex: first order =>EOS discontinuous
R. Balian
€
<E>=−∂β logZ
HIC03, Montreal 2003 1 C32
Energy Partition : E = Ek + Ep
<Ek> Thermometer
2Ck T2
Ck T22 C =
Abnormal fluctuations
Negativeheat capacity
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A robust signal2/T2
T
V=cte
p=cte
Out of equilibrium
Depends only on the state
Tsalis ensemble
With flow (up to 20%)
px
pzTransparency
x
zExpansion
HIC03, Montreal 2003 1 C32
Heat (Calories per grams)T
emp
erat
ure
(Deg
rees
)
Phase transition in infinite systems
Thermodynamical potentialsnon analytical
Reichl or Diu or …
€
F =T logZ( )
€
N→ ∞( )
Order of transition:discontinuity
Ehrenfest’s definition
Ex: first order:EOS discontinuous
R. Balian
€
<E>=−∂β logZ( )
€
∂β
nlogZ( )
HIC03, Montreal 2003 1 C32
1st order phase transition in finite systems
Zeroes of Z reach on real axis Yang & Lee Phys Rev 87(1952)404
Bimodal E distributionK.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
Ph. Ch. & F. Gulminelli Phys A (2003)
Back Bending in EOS (e.g. T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
EnergyTem
pera
ture
Re
Im
Complex
E distribution at
E1 E2
Energy
Caloric curve
HIC03, Montreal 2003 1 C32
1st order phase transition in finite systems
Zeroes of Z reach on real axis Yang & Lee Phys Rev 87(1952)404
Bimodal E distributionK.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
Ph. Ch. & F. Gulminelli Phys A (2003)
Back Bending in EOS (e.g. T(E))
Negative heat capacity K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
EnergyTem
pera
ture
Re
Im
Complex
E distribution at
E1 E2
Energy
Caloric curve
HIC03, Montreal 2003 1 C32
1st order in finite systems
Zeroes of Z reach on real axis Yang & Lee Phys Rev 87(1952)404
Bimodal E distributionK.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
PC & Gulminelli Phys A (2003)
Back Bending in EOS (e.g. T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Energy
Tem
pera
ture
Re
Im
Complex
E distribution at
E1 E2
Energy
Abnormal partial fluctuationJ.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
Energy k
/ T
2E1 E2
Caloric curve
Ck(3/2)
€
c
€
c
€
c
€
c
HIC03, Montreal 2003 1 C32
1st order phase transition in finite systems
Zeroes of Z in complex plane converge on real axis
Yang & Lee Phys Rev 87(1952)404
Bimodal E distributionorder parameters
K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
Ph. Ch. & F. Gulminelli Phys A (2003)
Back Bending in EOS
K.C. Lee Phys Rev E 53 (1952) 6558 Energy
Tem
pera
ture
Re
Im
Zeroes of Z()
E distribution at
E1 E2
HIC03, Montreal 2003 1 C32
Lattice-Gas Model
Closest neighbors Interaction
Metropolis Gibbs Equilibrium
Negative Susceptibility and Compressibility
Canonical
HIC03, Montreal 2003 1 C32
Generalization: inverted curvatures
Closest neighbors Interaction Metropolis Gibbs
Equilibrium Negative Susceptibility
and Compressibility
Canonical Lattice-Gas
Canonical Lattice-Gas
Particle Number
Pressure
Ch
emic
alP
oten
tial
Gulminelli, PC PRL99
HIC03, Montreal 2003 1 C32
Generalization: inverted curvatures
Closest neighbors interaction Metropolis Gibbs
equilibrium Negative susceptibility
and compressibility
Canonical Lattice-Gas
Canonical Lattice-Gas
Particle Number
Pressure
Ch
emic
alP
oten
tial
Gulminelli, PC PRL99
HIC03, Montreal 2003 1 C32
Density: Order parameter: Usually V=cst
(“box”) Negative
compressibility CV > 0
Lattice-Gas Model
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
HIC03, Montreal 2003 1 C32
1st order in finite systems
Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404
Bimodal E distribution (P (E)) K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
PC & Gulminelli Phys A (2003)
Back Bending in EOS (T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Energy
Tem
pera
ture
Re
Im
Complex
E distribution at
E1 E2
Energy
Abnormal fluctuation (k(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
Energy
k /
T2
E1 E2
Caloric curve
Ck(3/2)
€
c
€
c
€
c
€
c
Can
onic
alor
der
par
am. f
ree
Mic
roca
non
ical
ord
er p
aram
. fix
ed
HIC03, Montreal 2003 1 C32
Effects of V constraint
Constraint on the order parameter may suppress the negative heat capacity
Constraint on energy may suppress the bimodality
Isobar ensemble
P(i) exp-V(i)
HIC03, Montreal 2003 1 C32
Open systems (no box)Order param. bimodal
Bimodal P(V) Negative
compressibility Constraint on V
(order parameter) may suppress E bimodality CV > 0
Isobar ensemble
P(i) exp-V(i)
Isobarcanonicallattice gas
HIC03, Montreal 2003 1 C32
Volume distribution
Bimodal EventDistribution
Negative compressibility
Order parameter
Negative heat capacity
Isobar ensemble
P(i) exp-V(i)
HIC03, Montreal 2003 1 C32
1st order in finite systems
Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404
Bimodal E distribution (P (E)) K.C. Lee Phys Rev E 53 (1996) 6558
€
Pβ0 E( )
E1 E2
PC & Gulminelli Phys A (2003)
Back Bending in EOS (T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Energy
Tem
pera
ture
Re
Im
Complex
E distribution at
E1 E2
Energy
Abnormal fluctuation (k(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
Energy k
/ T
2E1 E2
Caloric curve
Ck(3/2)
€
c
€
c
€
c
€
c Order param. free (canonical)
Order param. fixe (microcanonical)
HIC03, Montreal 2003 1 C32
should we expect C<0 in multifragmentation?
energy is an order parameterenergy is an order parameter
C<0 C<0
with sharp boundary
without
Vorder
parameter
Energy
If V is notconstrained
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
Energy partition
E = Ek + Ep (kinetic+potential)
Wk(Ek)Wp(Ep)
W(E)
Sk(Ek) +S(Ep)-S(E)PE(Ek) = = e
Ck CP
T-2 (Ck+Cp)Fluctuations 2 : 2 =
Heat capacity Gaussian approximation lowest order
-1 -1Most probable Ek : Tk = ∂Esk = ∂ESp = Tp
Thermometer
Exact for classical gas
HIC03, Montreal 2003 1 C32
HIC03, Montreal 2003 1 C32
Energy Partition : E = Ek + Ep
<Ek> Thermometer
2Ck T2
Ck T22 C =
Abnormal fluctuations
Negativeheat capacity
HIC03, Montreal 2003 1 C32
A robust signal2/T2
T
V=cte
p=cte
Out of equilibrium
Depends only on the state
Tsalis ensemble
With flow (up to 20%)
px
pzTransparency
x
zExpansion
HIC03, Montreal 2003 1 C32
An observable: partial energy fluctuations
Canonical
Microcanonical
<Ek> = 3N T/2 classical
<Ek> =<iai>T2+3<m>T/2
Fermi
Ztot = ZkZp
E2k
2 +p2
2lnZk = Ck /k
2 lnZtot = C/ = E
2
Wtot = Wk Wp
E2 = 0
k2 =p
2
2lnWtot = -(Cf(k
2
2
1can
kk
C
C
−=
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