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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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phase transitions in nuclear reactions
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

slide5
I

-I-

Definition of phase transition

  • Discontinuity in derivatives of thermo potential when
  • 1st order equivalent to negative heat capacities when
phase transition in infinite systems
Phase transition in infinite systems

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)

1st order in finite systems
Complex b

Im(b)

b

Re(b)

  • Zeroes of Z reach real axis

Yang & Lee Phys Rev 87(1952)404

1st order in finite systems

PC & Gulminelli Phys A (2003)

  • Order param. free (canonical)
1st order in finite systems1
Complex b

Im(b)

b

Re(b)

b

E distribution at

  • Zeroes of Z reach real axis

Yang & Lee Phys Rev 87(1952)404

  • Bimodal E distribution(P(E))

Energy

K.C. Lee Phys Rev E 53 (1996) 6558

Caloric curve

b

Temperature

E1

E2

E1

E2

Energy

  • Back Bending in EOS(T(E))

K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

Ck(3/2)

  • Abnormal fluctuation (sk(E))

J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

E1

E2

Energy

1st order in finite systems

PC & Gulminelli Phys A (2003)

  • Order param. free (canonical)
  • Order param. fixe (microcanonical)

sk /T2

slide10
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
volume order parameter l g in a box v cst
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

Volume: order parameter L-G in a box: V=cst

Lattice-Gas Model

  • Negative compressibility

Gulminelli & PC PRL 82(1999)1402

volume order parameter l g in a box v cst1
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

Volume: order parameter L-G in a box: V=cst

Lattice-Gas Model

  • CV > 0
  • Negative compressibility

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.

open systems no box fluctuating volume
Isobar

canonical

lattice

gas

Isobar

ensemble

P(i) µexp-lV(i)

Open systems (no box)Fluctuating volume

PC, Duflot & Gulminelli PRL 85(2000)3587

  • Bimodal P(V)
    • Negative

compressibility

  • Bimodal P(E)
    • Negative heat

capacity Cp<0

microcanonical

lattice

gas

constrain on v i e on order paramameter
Isochore

canonical

lattice

gas

Constrain on Vi.e. on order paramameter
  • Suppresses bimodal P(E)
    • No negative heat capacity CV > 0

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.

c 0 in statistical models liquid gas and channel opening
Probability

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 modelsLiquid-gas and channel opening
    • q = 0 =>

no-Coulomb

  • Many channel opening
  • One Liquid-Gas transition

Bimodal P(EN)

slide17
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
coulomb reduces l g transition
Coulomb reduces L-G transition
  • Reduces C<0
    • Lattice-Gas
      • OK up to heavy nuclei
  • Reduces coexistence
    • Bonche-Levit-Vautherin

Nucl. Phys. A427 (1984) 278

A=207

Z= 82

Gulminelli, PC, Comment to PRC66 (2002) 041601

statistical treatment of non saturating forces interactions
Statistical treatment of Non saturating forces interactions

- Effective charge q = 1 charged system -

- Effective charge q = 0 uncharged system -

  • A unique framework: e-bE = e-bNEN -bCVC
      • Introduce two temperatures bN=band bC =q2b
      • O two energies ENand VC

bC = 0 Uncharged

bC = bNCharged

with and without coulomb a unique s e n v c
2

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

  • (EN ,VC) a unique phase diagram
  • Coulomb reduces E bimodality
    • different weight
    • rotation of E axis

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

correction of coulomb effects
0

0

2

2

4

4

6

6

8

8

10

10

12

12

14

14

Correction of Coulomb effects
  • If events overlap

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

partitions may differ for heavy nuclei
4

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
  • Channels are different

(cf fission)

  • Re-weighting impossible
  • However, a unique phase diagram

(EN VC)

Gulminelli, PC, Raduta, Raduta, submitted to PRL

radial flow at equilibrium
x

Thermal distribution in the moving frame

Flow induces

negative pressure

z

Expansion

Radial flow at “equilibrium”
  • Equilibrium = Max S under constrains
  • Additional constrains: radial flow
    • Additional Lagrange multiplier(r)
    • Self similar expansion = m a r =>(r) =  r
  • Requires a confining constrain => isobar ensemble

Gulminelli, PC, Nucl-Th (2002)

radial flow at equilibrium1
x

z

Expansion

Radial flow at “equilibrium”
  • Does no affect r partitioning
  • Only reduces the pressure
  • Shifts p distribution
  • Changes fragment distribution: fragmentation less effective
  • Requires a confining constrain => isobar ensemble

pressure

Gulminelli, PC, Nucl-Th (2002)

slide27
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
heat capacity from energy fluctuation
Canonical: total energy

Ztot = Zk Zp

sE2 = sk2 + sp2

2b log Zi = Ci /b2 = si2

  • Microcanonical: partial energy

Wtot = Wk  Wp

sE2 = 0, sk2 = sp2

2b log Wtot = -1/CT2 = f(sk2)

PC, Gulminelli NPA(1999)

Heat capacity from energy fluctuation
heat capacity from energy fluctuation1
Canonical: total energy

Ztot = Zk Zp

sE2 = sk2 + sp2

2b log Zi = Ci /b2 = si2

  • Microcanonical: partial energy

Wtot = Wk  Wp

sE2 = 0, sk2 = sp2

2b log Wtot = -1/CT2 = f(sk2)

PC, Gulminelli NPA(1999)

Heat capacity from energy fluctuation
a robust signal
2/T2

T

V=cte

p=cte

px

Tsalis ensemble

pz

Transparency

x

z

Expansion

A robust signal
  • Depends only on the state

PC, Duflot & Gulminelli PRL 85(2000)3587

  • Out of equilibrium
  • With flow (20%)

Gulminelli, PC Phisica A (2002)

Gulminelli, PC Nucl-th 2002

multifragmentation experiment

E

=

m

m

+

E

E

2

i

i

i

coul

coul

E

=E

-E

*

1

2

>=<

a

> T

2

+ 3/2 T

1

i

i

Multifragmentation experiment

Sort events in energy (Calorimetry)

Reconstruct a freeze-out partition

1- Primary fragments:

2- Freeze-out volume:

=> , 1

3- Kinetic EOS:

T, C1

heat capacity from energy fluctuation2
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

Heat capacity from energy fluctuation

Multics E1=20.3 E2=6.50.7

Isis E1=2.5 E2=7.

Indra E2=6.0.5

MULTICS

MULTICS

Excitation Energy (AMeV)

slide33
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

heat capacity from any fluctuation1
Heat capacity from any fluctuation
  • From kinetic energy
  • From any correlated observable (Ex. Abig)
  • r = skA /sksA correlation
  • sA “canonical” fluctuations
correction of experimental errors
Correction of experimental errors
  • Heat capacity from fluctuations
  • scan can be“filtered”
    • (apparatus and procedure)
correction of experimental errors1
Correction of experimental errors
  • Heat capacity from fluctuations
  • scan can be“filtered”
    • (apparatus and procedure)

Corrected

freeze-out

fluctuation

  • sk can be corrected
    • Iterate the procedure
      • Freeze-out reconstruction
      • Decay toward detector

Reconstructed

freeze-out

fluctuation

slide38
No Heat Bath

E  Etr

Order parameters: ,

Heat Bath

 = tr

= (SA30- SA2-30)/As

a alternative definition in the intensive ensemble
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 baxis perpendicularly

Yang Lee unit circle theorem

zero s of z
Zero’s of Z()

P1(E)

Partition sum

P2(b)

Monomodal distribution

Saddle point approximation

E1

K.C.Lee 1996, 2000

zero s of z1
Im(exp(t))

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

zero s of z2
Im(b)

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

slide57
IV

-3c-

Phase transition under flows

  • Example of oriented flow (transparency)
      • Exact thermodynamics
      • Effects on partitions
transparency at equilibrium
px

pz

Transparency

Transparency at “equilibrium”

{

t=-1 target

t =1 projectile

  • Equilibrium = Max S under constrains
  • Additional constrains: memory flow
    • Additional Lagrange multiplier
    • EOS leads to = Ap0with
  • Affects p space and fragments,
  • Does not affect the thermo if Eflow subtracted

Thermal distribution in moving frame

Gulminelli, PC, Nucl-Th (2002)

transparency at equilibrium1
px

pz

Transparency

Transparency at “equilibrium”
  • Does no affect r partitioning
  • Shifts p distribution
  • Changes fragment distribution: fragmentation less effective
  • Affects p space and fragments

Gulminelli, PC, Nucl-Th (2002)

transparency at equilibrium2
px

pz

Transparency

Transparency at “equilibrium”
  • Fragmentation less effective
  • Affect observables if Eflow not subtracted
    • Link between and
    • T from
    • Ex: Fluctuations of Q values
      • All thermal (from E = - 2e)
      • All relative motion
  • Up to 10-15% no effect
  • Affects p space and fragments

Gulminelli, PC, Nucl-Th (2002)

slide63
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
slide64
Order parameters: ,

Heat Bath

 = tr

No Heat Bath

E  Etr

= (SA30- SA2-30)/As

volume dependent t e
E T

V P

Volume dependent T(E)
  • 2-D EOS

Temperature

Pressure

Energy

volume dependent t e1
Lattice-Gas Model

E T

V P

Volume dependent T(E)
  • 2-D EOS

Temperature

Pressure

Energy

volume dependent t e2
Lattice-Gas Model

E T

V P

Volume dependent T(E)
  • 2-D EOS

Temperature

Critical point

First order

Liquid gas

Phase Transition

Pressure

Energy

volume dependent t e3
Lattice-Gas ModelVolume dependent T(E)

Transformation

dependent

Caloric Curve

- P = cst

- = cst

Temperature

  • Negative Heat Capacity

Pressure

Energy

volume dependent eos
Temperature

Fluctuations

Ck

Volume dependent EOS

P = cst

= cst

slide73
Ck

Volume dependent EOS

P = cst

= cst

slide74
P = cst

= cst

Volume dependent EOS

The Caloric curve

depends on the

transformation

C is not dT/dE

s2measures EOS

Abnormal in

Liquid-Gas

slide75
P = cst

= 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

liquid gas
Isobare

ensemble

P(i) µexp-lV(i)

Liquid-gas

Phase transition

and bimodality

liquid gas1
Isobare

ensemble

P(i) µexp-lV(i)

Liquid-gas

Phase transition

and bimodality

Negative heat

capacity

liquid gas2
Isobare

ensemble

P(i) µexp-lV(i)

Liquid-gas

Phase transition

And bimodality

Negative

compressibility

Negative heat

capacity

liquid gas3
Isobare

ensemble

P(i) µexp-lV(i)

Liquid-gas

Phase transition

And bimodality

Negative

compressibility

Negative heat

capacity

Order parameter

liquid gas4
Isobare

ensemble

P(i) µexp-lV(i)

Liquid-gas

Phase transition

And bimodality

Negative

compressibility

Negative heat

capacity

Order parameter

multifragmentation
Isobare

ensemble

P(i) µexp-lV(i)

Multifragmentation

Liquid

Gas

Statistical Model

(SMM-Raduta)

b = 3.7 MeV

A = 50

Z = 23

Bimodal

energy

distribution

Channel

openings

equilibrium what does it mean
Equilibrium : What Does It Mean?

Thermodynamics : one ∞ system

One ∞ system = ensemble of ∞ sub-systems

equilibrium what does it mean1
Equilibrium : What Does It Mean?

Finite system

Cannot be cut in sub-systems

equilibrium what does it mean2
Cannot be cut in sub-systemsEquilibrium : What Does It Mean?

One small system in time

if it is ergodic

Is a statistical ensemble

equilibrium what does it mean3
Equilibrium : What Does It Mean?

One small system in time

if it is ergodic

if it is chaotic

Many events

statistics from minimum information
Statistics from minimum information
  • “Chaos” populates phase space
statistics from minimum information1
Subtract

pre-

equili-

brium

Correct

for

evaporation

Statistics from minimum information
  • Compatible with freeze-out
  • “Chaos” populates phase space

BNV

statistics from minimum information2
Statistics from minimum information
  • “Chaos” populates phase space
statistics from minimum information3
Statistics from minimum information
  • Dynamics: global variables
  • Statistics: population
      • Many different ensembles
  • “Chaos” populates phase space

E

Microcanonical

Canonical

V

Isochore

Isobare

Deformed

Expanding

Grand

Rotating

...

Others

slide97
W equiprobable microstates

Entropy: missing information

  • T is the entropy increase

L. Boltzmann

What is temperature ?

T

R. Clausius

slide98
What is temperature ?
  • The microcanonical temperature

S =T-1 E

S = k logW

slide99
What is temperature ?

E = Ethermometer + Ebath

  • It is what thermometers measure.
  • The microcanonical temperature

S =T-1 E

S = k logW

slide100
What is temperature ?

E = Ethermometer + Ebath

  • It is what thermometers measure.

Distribution of microstates

  • The microcanonical temperature

S =T-1 E

S = k logW

slide101
P(Ethermo)

E

0

Ethermo

0

What is temperature ?

E = Ethermo + Ebath

  • It is what thermometers measure.

Distribution of microstates

P(Ethermo) Wthermo * Wbath

max P => logWthermo - logWbath=0

Most probable partition: Tthermo= bath

  • The microcanonical temperature

S =T-1 E

S = k logW

phase transition in infinite systems1
Temperature (Degrees)

Heat (Calories per grams)

Phase transition in infinite systems

Thermodynamical potentials

non analytical

Reichl or Diu or …

Order of transition:

discontinuity

Ehrenfest’s definition

Ex: first order:

EOS discontinuous

R. Balian

phase transition in infinite systems2
Phase transition in infinite systems

Thermodynamical potentials

non analytical

Reichl or Diu or …

Order of transition =>

discontinuous derivative

Ehrenfest’s definition

Ex: first order =>

EOS discontinuous

R. Balian

energy partition e e k e p
Ck T2

C =

Ck T2 - s2

Energy Partition : E = Ek + Ep

Thermometer

s2Abnormal

fluctuations

Negative

heat capacity

2

a robust signal1
2/T2

T

V=cte

p=cte

px

Tsalis ensemble

pz

Transparency

x

z

Expansion

A robust signal
  • Depends only on the state
  • Out of equilibrium
  • With flow (up to 20%)
phase transition in infinite systems3
Temperature (Degrees)

Heat (Calories per grams)

Phase transition in infinite systems

Thermodynamical potentials

non analytical

Reichl or Diu or …

Order of transition:

discontinuity

Ehrenfest’s definition

Ex: first order:

EOS discontinuous

R. Balian

1st order phase transition in finite systems
E1

E2

1st order phase transition in finite systems

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

1st order phase transition in finite systems1
E1

E2

1st order phase transition in finite systems

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

1st order in finite systems2
E1

E2

1st order in finite systems

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

1st order phase transition in finite systems2
Zeroes of Z(b)

Im(b)

b

Re(b)

E1

E2

1st order phase transition in finite systems

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

lattice gas model
Lattice-Gas Model

Canonical

  • Closest neighbors Interaction e
  • Metropolis Gibbs Equilibrium
  • Negative Susceptibility and Compressibility
generalization inverted curvatures
Generalization: inverted curvatures

Canonical Lattice-Gas

  • Closest neighbors Interaction e
    • Metropolis Gibbs Equilibrium
  • Negative Susceptibility and Compressibility

Chemical

Potential

Canonical Lattice-Gas

Gulminelli, PC

PRL99

Pressure

Particle Number

generalization inverted curvatures1
Generalization: inverted curvatures

Canonical Lattice-Gas

  • Closest neighbors interaction e
    • Metropolis Gibbs equilibrium
  • Negative susceptibility and compressibility

Chemical

Potential

Canonical Lattice-Gas

Gulminelli, PC

PRL99

Pressure

Particle Number

density order parameter
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

Density:Order parameter:
  • Usually V=cst
    • (“box”)
  • Negative compressibility
  • CV > 0

Canonical lattice gas

at constant V

1st order in finite systems3
E1

E2

1st order in finite systems

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

effects of v constraint
Isobar

ensemble

P(i) µexp-lV(i)

Effects of V constraint

Constraint on energy may suppress the bimodality

Constraint on the order parameter may suppress the negative heat capacity

open systems no box order param bimodal
Isobar

ensemble

P(i) µexp-lV(i)

Open systems (no box)Order param. bimodal
  • Bimodal P(V)
    • Negative compressibility
  • Constraint on V (order parameter) may suppress E bimodality
    • CV > 0

Isobar

canonical

lattice gas

volume distribution
Isobar

ensemble

P(i) µexp-lV(i)

Volume distribution

Bimodal Event

Distribution

Negative

compressibility

Negative heat

capacity

Order parameter

1st order in finite systems4
E1

E2

1st order in finite systems

Complex b

Im(b)

b

PC & Gulminelli Phys A (2003)

Re(b)

  • Order param. free (canonical)
  • Zeroes of Z reach real axis

b

E distribution at

Yang & Lee Phys Rev 87(1952)404

  • Bimodal E distribution(P(E))

Energy

K.C. Lee Phys Rev E 53 (1996) 6558

Caloric curve

b

Temperature

  • Order param. fixe (microcanonical)

E1

E2

Energy

  • Back Bending in EOS(T(E))

sk /T2

K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

Ck(3/2)

  • Abnormal fluctuation (sk(E))

J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

E1

E2

Energy

should we expect c 0 in multifragmentation
Energyshould we expect C<0 in multifragmentation?

r = A/V

order

parameter

with sharp

boundary

without

energy is an order parameter

C<0

If V is not

constrained

energy partition
Wk(Ek)Wp(Ep)

Sk(Ek) +S(Ep)-S(E)

PE(Ek) = = e

W(E)

-1

-1

  • Most probable Ek : Tk = ∂Esk = ∂ESp = Tp
    • Thermometer
  • Fluctuations s2 :s2 =
    • Heat capacity
    • Gaussian approximation lowest order

Ck CP

T-2 (Ck+Cp)

Energy partition
  • E = Ek + Ep (kinetic+potential)

Exact for classical gas

energy partition e e k e p1
Ck T2

C =

Ck T2 - s2

Energy Partition : E = Ek + Ep

Thermometer

s2Abnormal

fluctuations

Negative

heat capacity

2

a robust signal2
2/T2

T

V=cte

p=cte

px

Tsalis ensemble

pz

Transparency

x

z

Expansion

A robust signal
  • Depends only on the state
  • Out of equilibrium
  • With flow (up to 20%)
an observable partial energy fluctuations
An observable: partial energy fluctuations

= 3N T/2 classical

=<iai>T2+3T/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)

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