phase transitions in nuclear reactions

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≠∞


I

-I-Definition of phase transition

Discontinuity in derivatives of thermo potential when

1st order equivalent to negative heat capacities when

€

N→ ∞

€

N≠∞


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


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)


1st order in finite systems PC & Gulminelli Phys A (2003)


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

€


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


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

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


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

CV > 0

Lattice-Gas Model


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


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.


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


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


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)


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

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


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 -


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


Correction of Coulomb effects

If events overlap


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


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)



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∑


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


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


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

−=


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


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


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


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


Heat capacity from any fluctuation

2

2

1can

kk

C

C

−=

From kinetic energy


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


Correction of experimental errors

2

2

1can

kk

C

C

−=

Heat capacity from fluctuations

can can be “filtered”

(apparatus and procedure)


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


Heat Bath = tr

s

No Heat Bath E Etr

Order parameters: ,

€

Aasy= AA>Acut

∑ − AA≤Acut

∑

€

Abig


- 1b -Equivalence betweenbimodality of P(b) Zero's of the partition sum Z(b)


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


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


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

+


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


IV

-3c-Phase transition under flows

Example of oriented flow (transparency) Exact thermodynamics Effects on partitions


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



Affects p space and fragments


px

pzTransparency

Does no affect r partitioning

Shifts p distribution Changes fragment

distribution: fragmentation less effective



Affects p space and fragments


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>)


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


No Heat Bath E Etr

Heat Bath = tr

s

Order parameters: ,

€

Aasy= AA>Acut

∑ − AA≤Acut

∑

€

Abig


- 2b -Pressure dependent EOSCaloric Curves not EOS Fluctuations measure EOS


Volume dependent T(E)

Energy

Tem

per

atu

re

Pressu

re

2-D EOS

E TV P


Lattice-Gas Model


2-D EOS

Energy

Tem

per

atu

re

Pressu

re

E TV P


Lattice-Gas Model


2-D EOS

First orderLiquid gasPhase Transition

Critical point

Energy

Tem

per

atu

re

Pressu

re

E TV P


Lattice-Gas Model


Energy

Tem

per

atu

re

Pressu

re

Negative Heat Capacity

TransformationdependentCaloric Curve

- P = cst

- <V> = cst


Temperature

Ck

Fluctuations

Volume dependent EOS

<V> = cst

P = cst


Ck

P = cst <V> = cst Volume dependent EOS


Volume dependent EOS P = cst <V> = cst

The Caloric curvedepends on thetransformation

measures EOSAbnormal inLiquid-Gas

C is not dT/dE


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


Liquid-gas

Phase transitionand bimodality

Isobare ensemble

P(i) exp-V(i)


Liquid-gas

Phase transitionand bimodality

Negative heat capacity

Isobare ensemble

P(i) exp-V(i)


Liquid-gas

Phase transitionAnd bimodality



Isobare ensemble

P(i) exp-V(i)


Liquid-gas

Phase transitionAnd bimodality


Order parameter


Isobare ensemble

P(i) exp-V(i)


Multifragmentation

Statistical Model(SMM-Raduta)

Bimodalenergydistribution

Channelopenings

Isobare ensemble

P(i) exp-V(i)

= 3.7 MeVA = 50Z = 23

Liquid Gas


Equilibrium : What Does It Mean?

Thermodynamics : one ∞ system

One ∞ system = ensemble of ∞ sub-systems



Finite system

Cannot be cut in sub-systems


if it is ergodic


One small system in time

Cannot be cut in sub-systems

Is a statistical ensemble


if it is chaotic

if it is ergodic


One small system in time

Many events


Statistics from minimum information

“Chaos” populates phase space




Compatible with freeze-out

Correct forevaporation

Subtractpre-equili-brium

BNV


Dynamics: global variables Statistics: population

Many different ensembles



MicrocanonicalE <E> V <r3> <Q2> <p.r> <A> <L>

CanonicalIsochoreIsobareDeformed

ExpandingExpandingGrand

RotatingRotating

... Others


1. Temperature


What is temperature ?T

W equiprobable microstates

€

S=klogW

L. Boltzmann

Entropy: missing information

R. Clausius

T is the entropy increase

€

dS=dQT


What is temperature ?

The microcanonical temperatureS =T-1 ES = k logW



It is what thermometers measure.

E = Ethermometer + Ebath





E = Ethermometer + Ebath


Distribution of microstates


Most probable partition: Tthermo = bath


The microcanonical temperature


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


Heat (Calories per grams)T

emp

erat

ure

(Deg

rees

)



Reichl or Diu or …

€

F =T logZ( )

€

N→ ∞( )




R. Balian

€


€

∂β

nlogZ( )





€

F =T logZ

€

N→ ∞

Order of transition =>discontinuous derivative


Ex: first order =>EOS discontinuous

R. Balian

€

<E>=−∂β logZ


Energy Partition : E = Ek + Ep

<Ek> Thermometer

2Ck T2

Ck T22 C =

Abnormal fluctuations

Negativeheat capacity


A robust signal2/T2

T

V=cte

p=cte

Out of equilibrium


Tsalis ensemble

With flow (up to 20%)

px

pzTransparency

x

zExpansion


Heat (Calories per grams)T

emp

erat

ure

(Deg

rees

)




€

F =T logZ( )

€

N→ ∞( )




R. Balian

€


€

∂β

nlogZ( )


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





€

Pβ0 E( )

E1 E2


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


1st order in finite systems



€

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



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


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


Lattice-Gas Model

Closest neighbors Interaction

Metropolis Gibbs Equilibrium

Negative Susceptibility and Compressibility

Canonical


Generalization: inverted curvatures

Closest neighbors Interaction Metropolis Gibbs

Equilibrium Negative Susceptibility

and Compressibility

Canonical Lattice-Gas


Particle Number

Pressure

Ch

emic

alP

oten

tial

Gulminelli, PC PRL99


Generalization: inverted curvatures

Closest neighbors interaction Metropolis Gibbs

equilibrium Negative susceptibility

and compressibility



Particle Number

Pressure

Ch

emic

alP

oten

tial

Gulminelli, PC PRL99


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






€

Pβ0 E( )

E1 E2


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


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


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)


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


Volume distribution

Bimodal EventDistribution


Order parameter


Isobar ensemble

P(i) exp-V(i)





€

Pβ0 E( )

E1 E2


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


Energy k

/ T

2E1 E2

Caloric curve

Ck(3/2)

€

c

€

c

€

c

€


Order param. fixe (microcanonical)


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


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


Energy Partition : E = Ek + Ep

<Ek> Thermometer

2Ck T2

Ck T22 C =

Abnormal fluctuations

Negativeheat capacity


A robust signal2/T2

T

V=cte

p=cte

Out of equilibrium


Tsalis ensemble

With flow (up to 20%)

px

pzTransparency

x

zExpansion


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

−=

phase transitions in nuclear reactions

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