Phase Transitions in Nuclear Reactions. Definition of phase transitions Non analytical thermo potential Negative heat capacity (1 st order) Liquid gas case C<0 and Volume (order parameter) fluctuations Thermo of nuclear reactions Coulomb reduces coexistence and C<0 region
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Ph. Chomaz and F. Gulminelli, Caen France
I
Definition of phase transition
Thermodynamical potentials
non analytical
L.E. Reichl, Texas Press (1980)
Caloric curve
Order of transition:
discontinuity
b
Temperature
Ehrenfest’s definition
E1
E2
Energy
Ex: first order:
EOS discontinuous
R. Balian, Springer (1982)
Im(b)
b
Re(b)
Yang & Lee Phys Rev 87(1952)404
1st order in finite systemsPC & Gulminelli Phys A (2003)
Im(b)
b
Re(b)
b
E distribution at
Yang & Lee Phys Rev 87(1952)404
Energy
K.C. Lee Phys Rev E 53 (1996) 6558
Caloric curve
b
Temperature
E1
E2
E1
E2
Energy
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Ck(3/2)
J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
E1
E2
Energy
1st order in finite systemsPC & Gulminelli Phys A (2003)
sk /T2
II
Liquid gas transition
at constant V
T=10
MeV
8
Pressure ( MeV/fm3)
4 2 0 2 4 6 8
6
4
0 0.2
0.4 0.6
0.8
r / r
Density
0
Volume: order parameter LG in a box: V=cstLatticeGas Model
Gulminelli & PC PRL 82(1999)1402
at constant V
T=10
MeV
8
Pressure ( MeV/fm3)
Thermo
limit
4 2 0 2 4 6 8
6
4
Thermo limit
0 0.2
0.4 0.6
0.8
r / r
Density
0
Volume: order parameter LG in a box: V=cstLatticeGas Model
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., condmat/0302177, Phys. Rev. E.
canonical
lattice
gas
Isobar
ensemble
P(i) µexplV(i)
Open systems (no box)Fluctuating volumePC, Duflot & Gulminelli PRL 85(2000)3587
compressibility
capacity Cp<0
microcanonical
lattice
gas
canonical
lattice
gas
Constrain on Vi.e. on order paramametermicrocanonical
lattice
gas
See specific discussion for large systems
Pleimling and Hueller, J.Stat.Phys.104 (2001) 971
Binder, Physica A 319 (2002) 99.
Gulminelli et al., condmat/0302177, Phys. Rev. E.
2
Coulomb interaction VC
1.6
1.2
0.8
0.4
Probability
0
2
4
6
8
10
12
14
Nuclear energy
C<0 in statistical modelsLiquidgas and channel openingnoCoulomb
Bimodal P(EN)
III
Thermo of nuclear reaction
Nucl. Phys. A427 (1984) 278
A=207
Z= 82
Gulminelli, PC, Comment to PRC66 (2002) 041601
 Effective charge q = 1 charged system 
 Effective charge q = 0 uncharged system 
bC = 0 Uncharged
bC = bNCharged
1.6
1.2
0.8
0.4
0
2
4
6
8
10
12
14
With and without Coulomb a unique S(EN,VC)Probability
0
4
8
12
Energy
Coulomb interaction VC
bC = Uncharged
bC = bNCharged
E=EN+VC, charged
E=EN, uncharged
Energy
Gulminelli, PC, Raduta, Raduta, submitted to PRL
0
2
2
4
4
6
6
8
8
10
10
12
12
14
14
Correction of Coulomb effectsEnergy distribution: Entropy
Reconstructed
Exact
Probability
2
2
1.6
1.6
bC = 0
Uncharged
Coulomb interaction VC
Coulomb interaction VC
1.2
1.2
bC = bN
Charged
Reconstruction
of the uncharged distribution from the charged one
0.8
0.8
Energy
Energy
Gulminelli, PC, Raduta, Raduta, submitted to PRL
bC = 0
Uncharged
3
Coulomb interaction VC
2
1
bC = bNCharged
0
2
4
6
8
10
12
14
Energy
Partitions may differ for heavy nuclei(cf fission)
(EN VC)
Gulminelli, PC, Raduta, Raduta, submitted to PRL
Thermal distribution in the moving frame
Flow induces
negative pressure
z
Expansion
Radial flow at “equilibrium”Gulminelli, PC, NuclTh (2002)
z
Expansion
Radial flow at “equilibrium”pressure
Gulminelli, PC, NuclTh (2002)
IV
Conclusion and application to data
Ztot = Zk Zp
sE2 = sk2 + sp2
2b log Zi = Ci /b2 = si2
Wtot = Wk Wp
sE2 = 0, sk2 = sp2
2b log Wtot = 1/CT2 = f(sk2)
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuationZtot = Zk Zp
sE2 = sk2 + sp2
2b log Zi = Ci /b2 = si2
Wtot = Wk Wp
sE2 = 0, sk2 = sp2
2b log Wtot = 1/CT2 = f(sk2)
PC, Gulminelli NPA(1999)
Heat capacity from energy fluctuationT
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
A robust signalPC, Duflot & Gulminelli PRL 85(2000)3587
Gulminelli, PC Phisica A (2002)
Gulminelli, PC Nuclth 2002
E
=
m
m
+
E
E
•
2
i
i
i
coul
coul
E
=E
E
*
•
1
2
•
>=<
a
> T
2
+ 3/2
1
i
i
Multifragmentation experimentSort events in energy (Calorimetry)
Reconstruct a freezeout partition
1 Primary fragments:
2 Freezeout volume:
=>
3 Kinetic EOS:
T, C1
2
D’Agostino et al.
Hot Au, ISIS collaboration
Fluctuation, Heat Capacity
1
INDRA
0
0
2
4
6
8
Excitation Energy (AMeV)
Bougault et al Xe+Sn central
ISIS
ISIS
INDRA
Heat capacity from energy fluctuationMultics E1=20.3 E2=6.50.7
Isis E1=2.5 E2=7.
Indra E2=6.0.5
MULTICS
MULTICS
Excitation Energy (AMeV)
Au+Au 35 EF=1
Comparisoncentral / peripheral
peripheral
M.D ’AgostinoIWM2001
central
Ckin=
dEkin/dT
Up to 35 A.MeV the flow ambiguity
is a small effect
Corrected
freezeout
fluctuation
Reconstructed
freezeout
fluctuation
is defined by
Yang Lee unit circle theorem
P1(E)
Partition sum
P2(b)
Monomodal distribution
Saddle point approximation
E1
K.C.Lee 1996, 2000
Re(exp(t))
Im(b)
Re(b)
Zero’s of Z()+
P1(E)
P2(E)
Partition sum
Bimodal distribution
E1
E2
Saddle point approximation
b
K.C.Lee 1996, 2000
Re(b)
Zero’s of Z()
Factorize the N zero’s []
b
Inverse Laplace
2 distributions splitted by the latent heat N
Ph. Ch., F. Gulminelli, 2003
3c
Phase transition under flows
pz
Transparency
Transparency at “equilibrium”{
t=1 target
t =1 projectile
Thermal distribution in moving frame
Gulminelli, PC, NuclTh (2002)
pz
Transparency
Transparency at “equilibrium”Gulminelli, PC, NuclTh (2002)
pz
Transparency
Transparency at “equilibrium”Gulminelli, PC, NuclTh (2002)
V
Phase transition other signal
E T
V P
Volume dependent T(E)Temperature
Critical point
First order
Liquid gas
Phase Transition
Pressure
Energy
Transformation
dependent
Caloric Curve
 P = cst

Temperature
Pressure
Energy
Volume dependent EOS
The Caloric curve
depends on the
transformation
C is not dT/dE
s2measures EOS
Abnormal in
LiquidGas
Caloric curve are not EOSFluctuations are
The Caloric curve
depends on the
transformation
C is not dT/dE
s2measures EOS
s2abnormal in
LiquidGas
ensemble
P(i) µexplV(i)
LiquidgasPhase transition
And bimodality
Negative
compressibility
Negative heat
capacity
ensemble
P(i) µexplV(i)
LiquidgasPhase transition
And bimodality
Negative
compressibility
Negative heat
capacity
Order parameter
ensemble
P(i) µexplV(i)
LiquidgasPhase transition
And bimodality
Negative
compressibility
Negative heat
capacity
Order parameter
ensemble
P(i) µexplV(i)
MultifragmentationLiquid
Gas
Statistical Model
(SMMRaduta)
b = 3.7 MeV
A = 50
Z = 23
Bimodal
energy
distribution
Channel
openings
Thermodynamics : one ∞ system
One ∞ system = ensemble of ∞ subsystems
One small system in time
if it is ergodic
Is a statistical ensemble
pre
equili
brium
Correct
for
evaporation
Statistics from minimum informationBNV
E
Microcanonical
Canonical
V
Isochore
Isobare
Deformed
Expanding
Entropy: missing information
L. Boltzmann
What is temperature ?
TR. Clausius
E = Ethermometer + Ebath
S =T1 E
S = k logW
E = Ethermometer + Ebath
Distribution of microstates
S =T1 E
S = k logW
E
0
Ethermo
0
What is temperature ?
E = Ethermo + Ebath
Distribution of microstates
P(Ethermo) Wthermo * Wbath
max P => logWthermo  logWbath=0
Most probable partition: Tthermo= bath
S =T1 E
S = k logW
Heat (Calories per grams)
Phase transition in infinite systemsThermodynamical potentials
non analytical
Reichl or Diu or …
Order of transition:
discontinuity
Ehrenfest’s definition
Ex: first order:
EOS discontinuous
R. Balian
Thermodynamical potentials
non analytical
Reichl or Diu or …
Order of transition =>
discontinuous derivative
Ehrenfest’s definition
Ex: first order =>
EOS discontinuous
R. Balian
C =
Ck T2  s2
Energy Partition : E = Ek + Eps2Abnormal
fluctuations
Negative
heat capacity
2
T
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
A robust signalHeat (Calories per grams)
Phase transition in infinite systemsThermodynamical potentials
non analytical
Reichl or Diu or …
Order of transition:
discontinuity
Ehrenfest’s definition
Ex: first order:
EOS discontinuous
R. Balian
E2
1st order phase transition in finite systemsPh. Ch. & F. Gulminelli Phys A (2003)
Complex b
Im(b)
b
Zeroes of Z reach on real axis
Re(b)
Yang & Lee Phys Rev 87(1952)404
b
E distribution at
Bimodal E distribution
K.C. Lee Phys Rev E 53 (1996) 6558
Energy
Back Bending in EOS(e.g. T(E))
Caloric curve
b
Temperature
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
E1
E2
Energy
E2
1st order phase transition in finite systemsPh. Ch. & F. Gulminelli Phys A (2003)
Complex b
Im(b)
b
Zeroes of Z reach on real axis
Re(b)
Yang & Lee Phys Rev 87(1952)404
b
E distribution at
Bimodal E distribution
K.C. Lee Phys Rev E 53 (1996) 6558
Energy
Back Bending in EOS(e.g. T(E))
Negative heat capacity
Caloric curve
b
Temperature
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
E1
E2
Energy
E2
1st order in finite systemsComplex b
Im(b)
b
PC & Gulminelli Phys A (2003)
Re(b)
Zeroes of Z reach on real axis
Yang & Lee Phys Rev 87(1952)404
b
E distribution at
Bimodal E distribution
K.C. Lee Phys Rev E 53 (1996) 6558
Energy
Back Bending in EOS(e.g. T(E))
Caloric curve
b
Temperature
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
E1
E2
Energy
Abnormal partial fluctuation
sk /T2
Ck(3/2)
J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
E1
E2
Energy
Im(b)
b
Re(b)
E1
E2
1st order phase transition in finite systemsPh. Ch. & F. Gulminelli Phys A (2003)
Zeroes of Z in complex b
plane converge on real axis
Yang & Lee Phys Rev 87(1952)404
b
Bimodal E distribution
order parameters
E distribution at
K.C. Lee Phys Rev E 53 (1996) 6558
b
Back Bending in EOS
Temperature
E1
E2
Energy
K.C. Lee Phys Rev E 53 (1952) 6558
Canonical
Canonical LatticeGas
Chemical
Potential
Canonical LatticeGas
Gulminelli, PC
PRL99
Pressure
Particle Number
Canonical LatticeGas
Chemical
Potential
Canonical LatticeGas
Gulminelli, PC
PRL99
Pressure
Particle Number
T=10
MeV
8
Pressure ( MeV/fm3)
4 2 0 2 4 6 8
6
4
0.4 0.6
0.8
0 0.2
r / r
Density
0
Density:Order parameter:Canonical lattice gas
at constant V
E2
1st order in finite systemsComplex b
Im(b)
b
PC & Gulminelli Phys A (2003)
Re(b)
Zeroes of Z reach real axis
Yang & Lee Phys Rev 87(1952)404
Canonical
order param. free
b
E distribution at
Bimodal E distribution(P(E))
K.C. Lee Phys Rev E 53 (1996) 6558
Energy
Back Bending in EOS(T(E))
Caloric curve
b
Temperature
Microcanonical
order param. fixed
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
E1
E2
Energy
Abnormal fluctuation (sk(E))
sk /T2
Ck(3/2)
J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
E1
E2
Energy
ensemble
P(i) µexplV(i)
Effects of V constraintConstraint on energy may suppress the bimodality
Constraint on the order parameter may suppress the negative heat capacity
ensemble
P(i) µexplV(i)
Open systems (no box)Order param. bimodalIsobar
canonical
lattice gas
ensemble
P(i) µexplV(i)
Volume distributionBimodal Event
Distribution
Negative
compressibility
Negative heat
capacity
Order parameter
E2
1st order in finite systemsComplex b
Im(b)
b
PC & Gulminelli Phys A (2003)
Re(b)
b
E distribution at
Yang & Lee Phys Rev 87(1952)404
Energy
K.C. Lee Phys Rev E 53 (1996) 6558
Caloric curve
b
Temperature
E1
E2
Energy
sk /T2
K. Binder, D.P. Landau Phys Rev B30 (1984) 1477
Ck(3/2)
J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153
E1
E2
Energy
r = A/V
order
parameter
with sharp
boundary
without
energy is an order parameter
C<0
If V is not
constrained
Sk(Ek) +S(Ep)S(E)
PE(Ek) = = e
W(E)
1
1
Ck CP
T2 (Ck+Cp)
Energy partitionExact for classical gas
C =
Ck T2  s2
Energy Partition : E = Ek + Eps2Abnormal
fluctuations
Negative
heat capacity
2
T
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
A robust signalFermi
Canonical
Ztot = ZkZp
sE2 = sk2 + sp2
2b lnZk = Ck /b2 = sk2
2b lnZtot = C/b2 = sE2
Microcanonical
Wtot = Wk Wp
sE2 = 0
sk2 = sp2
2b lnWtot = (CT2)1 = f(sk2)