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)
Complex b
Im(b)
b
Re(b)
Yang & Lee Phys Rev 87(1952)404
PC & Gulminelli Phys A (2003)
Complex b
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
PC & Gulminelli Phys A (2003)
sk /T2
-II-
Liquid gas transition
Canonical lattice gas
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
Lattice-Gas Model
Gulminelli & PC PRL 82(1999)1402
Canonical lattice gas
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
Lattice-Gas 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., cond-mat/0302177, Phys. Rev. E.
Isobar
canonical
lattice
gas
Isobar
ensemble
P(i) µexp-lV(i)
PC, Duflot & Gulminelli PRL 85(2000)3587
compressibility
capacity Cp<0
microcanonical
lattice
gas
Isochore
canonical
lattice
gas
microcanonical
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., cond-mat/0302177, Phys. Rev. E.
Probability
2
Coulomb interaction VC
1.6
1.2
0.8
0.4
Probability
0
2
4
6
8
10
12
14
Nuclear energy
no-Coulomb
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
2
1.6
1.2
0.8
0.4
0
2
4
6
8
10
12
14
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
0
2
2
4
4
6
6
8
8
10
10
12
12
14
14
Energy 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
4
bC = 0
Uncharged
3
Coulomb interaction VC
2
1
bC = bNCharged
0
2
4
6
8
10
12
14
Energy
(cf fission)
(EN VC)
Gulminelli, PC, Raduta, Raduta, submitted to PRL
x
Thermal distribution in the moving frame
Flow induces
negative pressure
z
Expansion
Gulminelli, PC, Nucl-Th (2002)
x
z
Expansion
pressure
Gulminelli, PC, Nucl-Th (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)
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)
2/T2
T
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
PC, Duflot & Gulminelli PRL 85(2000)3587
Gulminelli, PC Phisica A (2002)
Gulminelli, PC Nucl-th 2002
E
=
m
m
+
E
E
•
2
i
i
i
coul
coul
E
=E
-E
*
•
1
2
•
<E
>=<
a
> T
2
+ 3/2 <M-1> T
1
i
i
Sort events in energy (Calorimetry)
Reconstruct a freeze-out partition
1- Primary fragments:
2- Freeze-out volume:
=> <E1>, 1
3- Kinetic EOS:
T, C1
Ni+Ni, 32 AMeV Quasi-Projectile
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
Multics 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=0
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
freeze-out
fluctuation
Reconstructed
freeze-out
fluctuation
No Heat Bath
E Etr
Order parameters: ,
Heat Bath
= tr
= (SA30- SA2-30)/As
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
Im(exp(t))
Re(exp(t))
Im(b)
Re(b)
+
P1(E)
P2(E)
Partition sum
Bimodal distribution
E1
E2
Saddle point approximation
b
K.C.Lee 1996, 2000
Im(b)
Re(b)
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
px
pz
Transparency
{
t=-1 target
t =1 projectile
Thermal distribution in moving frame
Gulminelli, PC, Nucl-Th (2002)
px
pz
Transparency
Gulminelli, PC, Nucl-Th (2002)
px
pz
Transparency
Gulminelli, PC, Nucl-Th (2002)
-V-
Phase transition other signal
Order parameters: ,
Heat Bath
= tr
No Heat Bath
E Etr
= (SA30- SA2-30)/As
E T
V P
Temperature
Pressure
Energy
Lattice-Gas Model
E T
V P
Temperature
Pressure
Energy
Lattice-Gas Model
E T
V P
Temperature
Critical point
First order
Liquid gas
Phase Transition
Pressure
Energy
Lattice-Gas Model
Transformation
dependent
Caloric Curve
- P = cst
- <V> = cst
Temperature
Pressure
Energy
Temperature
Fluctuations
Ck
P = cst
<V> = cst
Ck
Volume dependent EOS
P = cst
<V> = cst
P = cst
<V> = cst
Volume dependent EOS
The Caloric curve
depends on the
transformation
C is not dT/dE
s2measures EOS
Abnormal in
Liquid-Gas
P = cst
<V> = cst
Caloric curve are not EOSFluctuations are
The Caloric curve
depends on the
transformation
C is not dT/dE
s2measures EOS
s2abnormal in
Liquid-Gas
Isobare
ensemble
P(i) µexp-lV(i)
Phase transition
and bimodality
Isobare
ensemble
P(i) µexp-lV(i)
Phase transition
and bimodality
Negative heat
capacity
Isobare
ensemble
P(i) µexp-lV(i)
Phase transition
And bimodality
Negative
compressibility
Negative heat
capacity
Isobare
ensemble
P(i) µexp-lV(i)
Phase transition
And bimodality
Negative
compressibility
Negative heat
capacity
Order parameter
Isobare
ensemble
P(i) µexp-lV(i)
Phase transition
And bimodality
Negative
compressibility
Negative heat
capacity
Order parameter
Isobare
ensemble
P(i) µexp-lV(i)
Liquid
Gas
Statistical Model
(SMM-Raduta)
b = 3.7 MeV
A = 50
Z = 23
Bimodal
energy
distribution
Channel
openings
Thermodynamics : one ∞ system
One ∞ system = ensemble of ∞ sub-systems
Finite system
Cannot be cut in sub-systems
Cannot be cut in sub-systems
One small system in time
if it is ergodic
Is a statistical ensemble
One small system in time
if it is ergodic
if it is chaotic
Many events
Subtract
pre-
equili-
brium
Correct
for
evaporation
BNV
E
Microcanonical
<E>
Canonical
V
Isochore
<r3>
Isobare
<Q2>
Deformed
<p.r>
Expanding
<A>
Grand
<L>
Rotating
...
Others
1. Temperature
Entropy: missing information
L. Boltzmann
What is temperature ?
R. Clausius
What is temperature ?
S =T-1 E
S = k logW
What is temperature ?
E = Ethermometer + Ebath
S =T-1 E
S = k logW
What is temperature ?
E = Ethermometer + Ebath
Distribution of microstates
S =T-1 E
S = k logW
P(Ethermo)
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 =T-1 E
S = k logW
Temperature (Degrees)
Heat (Calories per grams)
Thermodynamical 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
Ck T2
C =
Ck T2 - s2
<Ek> Thermometer
s2Abnormal
fluctuations
Negative
heat capacity
2
2/T2
T
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
Temperature (Degrees)
Heat (Calories per grams)
Thermodynamical potentials
non analytical
Reichl or Diu or …
Order of transition:
discontinuity
Ehrenfest’s definition
Ex: first order:
EOS discontinuous
R. Balian
E1
E2
Ph. 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
E1
E2
Ph. 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
E1
E2
Complex 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
Zeroes of Z(b)
Im(b)
b
Re(b)
E1
E2
Ph. 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 Lattice-Gas
Chemical
Potential
Canonical Lattice-Gas
Gulminelli, PC
PRL99
Pressure
Particle Number
Canonical Lattice-Gas
Chemical
Potential
Canonical Lattice-Gas
Gulminelli, PC
PRL99
Pressure
Particle Number
Lattice-Gas Model
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
Canonical lattice gas
at constant V
E1
E2
Complex 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
Isobar
ensemble
P(i) µexp-lV(i)
Constraint on energy may suppress the bimodality
Constraint on the order parameter may suppress the negative heat capacity
Isobar
ensemble
P(i) µexp-lV(i)
Isobar
canonical
lattice gas
Isobar
ensemble
P(i) µexp-lV(i)
Bimodal Event
Distribution
Negative
compressibility
Negative heat
capacity
Order parameter
E1
E2
Complex 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
Energy
r = A/V
order
parameter
with sharp
boundary
without
energy is an order parameter
C<0
If V is not
constrained
Wk(Ek)Wp(Ep)
Sk(Ek) +S(Ep)-S(E)
PE(Ek) = = e
W(E)
-1
-1
Ck CP
T-2 (Ck+Cp)
Exact for classical gas
Ck T2
C =
Ck T2 - s2
<Ek> Thermometer
s2Abnormal
fluctuations
Negative
heat capacity
2
2/T2
T
V=cte
p=cte
px
Tsalis ensemble
pz
Transparency
x
z
Expansion
<Ek> = 3N T/2 classical
<Ek> =<iai>T2+3<m>T/2
Fermi
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)