phase transitions in nuclear reactions
DESCRIPTION
Phase Transitions in Nuclear Reactions. Definition of phase transitions Non analytical thermo potential Negative heat capacity (1 st order) Liquid gas case CTRANSCRIPT
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≠∞
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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
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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
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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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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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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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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
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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
−=
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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
−=
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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
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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
−=
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)
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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) 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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- 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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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
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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 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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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
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- 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
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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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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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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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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
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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>
CanonicalIsochoreIsobareDeformed
ExpandingExpandingGrand
RotatingRotating
... Others
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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
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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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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( )
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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
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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
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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( )
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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
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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
−=