Phase transition in finite systems

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Phase transition in finite systems

Philippe CHOMAZ, GANIL

Finite systems Zeroes of the partition sum Bimodal event distributions Negative heat capacity Abnormal fluctuations

Experimental results Melting of Na clusters Superfluidity in atomic nuclei Fragmentation of nuclei Fragmentation of H-clusters

2

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*****Chapitre 1

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- I -Introduction

Phase transitionAnomaly in thermo

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EOS:

€

<E>=−∂β logZ

Ex: first order:discontinuous

Phase transition in infinite systems

R. Balian, Springer (1982)

Order n:discontinuity in

Ehrenfest’s definition

€

∂β

nlogZ

Caloric curve

€

Z = e−βE ( n)

(n )∑ ⎛

⎝ ⎜ ⎞

⎠ ⎟

L.E. Reichl, Texas Press (1980)

€

N→ ∞Thermodynamical potentials:

€

e.g. F = −T logZ

€

βEOS

Temperature-1

Ene

rgy

<E

>

Thermodynamical potentials:non analytical at

4

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No phase transition continuous


EOS:

€

<E>=−∂β logZ

Ex: Continuous

€

N→ ∞

€

βEOS

Temperature-1

Ene

rgy

<E

>

€

≠Caloric curve

Thermodynamical potentials: Z >0,

€

∂β

nlogZ


€

Z = e−βE ( n)

(n )∑ ⎛

⎝ ⎜ ⎞

⎠ ⎟

€

e.g. F = −T logZ

6

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But anomaly in the entropy Ex: negative heat capacity



No phase transition continuous


€

N→ ∞

€

≠

€

βEOS

Temperature-1

Ene

rgy

<E

>

Caloric curve

Thermodynamical potentials: Z >0,

€

∂β

nlogZ


€

Z = e−βE ( n)

(n )∑ ⎛

⎝ ⎜ ⎞

⎠ ⎟

€

e.g. F = −T logZ

EnergyTem

pera

ture

E1 E2

Caloric curve

K. Binder, D.P. Landau Phys Rev B30 (1984) 1477; Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495.; W. Thirring, Z. Phys. 235, 339 (1970), D.H.E. Gross, Rep. Prog. Phys. 53, 605 (1990),

€

C−1 = ∂ET€

T−1 = ∂E S

8

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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) 1477Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495.

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 Free order param. (canonical)

Fixed order para. (microcanonical)

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

€

Pβ0 E( )

E1 E2

E distribution at

Energy

10

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*****Chapitre 2

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- II -Origin of singularities

Zeroes of Z ()

Bimodal P(E)

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, then logZ remains analytical=> No phase transition

Discontinuities from zeroes of Z

Re zeroes of Z

Im

Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z

(C.N.Yang T.D.Lee 1952)

If, when , no zeroes of Z converge on the Real axis

€

N→∞

Complex plane

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, then logZ remains analytical=> No phase transition

Discontinuities from zeroes of Z

Re zeroes of Z

Im

Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z

(C.N.Yang T.D.Lee 1952)

If, when , no zeroes of Z converge on the Real axis

€

N→∞

Complex plane

0 First order phase transitions

A uniform density of zero's on a line crossing the real axis perpendicularly at 0

Im

Re

Yang - Lee theorem

12

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Im

Re

Yang - Lee theorem

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A theoretical definition Only valid asymptotically: V = How to extend it, V ≠ ?

How to use it experimentally ?



Im

Re

Yang - Lee theorem

∞∞

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Im

Re

Yang - Lee theorem

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First order phase transitions A uniform density of zero's on a line crossing the real axis perpendicularly at 0

Link with energy distribution P(E)

0Im

Re

€

Zβ( )=Zβ0( ) dEPβ0 E( )e−β0−β( )E∫

Z Laplace transform of P:

€

Pβ E( )=W(E)e−βE/Zβ( )

Ph. Ch., F. Gulminelli, Physica A 2003

€

Zβ( )= dEW E( )e−βE∫

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First order phase transitions A uniform density of zero's on a line crossing the real axis perpendicularly at 0

Link with energy distribution P(E)

0Im

Re

P1(E)

P2(E)

Bimodal P with EN

Z Laplace transform of P:

€

Pβ E( )=W(E)e−βE/Zβ( )

<=

> Ph. Ch., F. Gulminelli, Physica A 2003

€

Zβ( )= dEW E( )e−βE∫

€

Zβ( )=Zβ0( ) dEPβ0 E( )e+β0−β( )E∫

14

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Fin Zero=>bimodalite

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Closest neighbors interaction

Metropolis Boltzmann weight e-E

Average volume <V> (pressure)

Lattice gas model

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Canonical(Average)

Lattice-gas Model




Lattice gas model

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

<E> smooth fonction of (canonical caloric curve)

16

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Canonical(Average)

Lattice-gas Model




Lattice gas model

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

<E> smooth function of (canonical caloric curve)

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Canonical(Average)

Lattice-gas Model

En

erg

y D

istr

ibu

tio

n

1

10

100

0.1

Liquid Gas

Energy distribution at a fix temperature

Canonical(Most Probable)

Lattice gas model

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

18

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Canonical(Average)


Lattice-gas Model

En

erg

y D

istr

ibu

tio

n d

1

10

100

0.1

Liquid Gas

Energy distribution at a fixed temperature

Discontinuity of the most probable

Lattice-gas model

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

18

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Canonical(Average)


Lattice-gas Model

Dis

trib

uti

on

d ’

éne

rgie

1

10

100

0.1



Liquid GasLattice-gas model

Bimodal distribution

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

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A first order phase transition in finite systems

A bimodal (event) distribution of energy

If the energy is free to fluctuate (canonical - energy reservoire)

<=

>

20

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A bimodal (event) distribution of an observable (order parameter)

If this observable is free to fluctuate (intensive ensemble)

<=

>

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Access to order parameters Nature of order parameter:

Is it collective? Scaling of the jump with N:

What happen at the thermo limit (N = ∞) ?

Is it a macro. phase transition?

A bimodal (event) distribution of an observable (order parameter)

If this observable is free to fluctuate (intensive ensemble)

A1

A2

A3

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Tem

per

atur

e,

Pro

bab

ilit

y P

(E

)

Energy

Energy distribution

Photo-dissociation

3h 4h 5h

Number of evaporated atoms

Haberland-PRL 2001

Melting of Na Cluster

Liquid-gas in lattice-gas

M

Ising model

Fragment asymmetry (Zbig-Zsmall)

INDRA

Multifragmentation of nuclei

INDRA-2001

Bimodal distributions

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***** bébut Na

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Melting of Na147 .

++

energy

P(E

)

M.Schmidt et al, Nature 1999, PRL 2001

Variable THelium

Heat Bath

E*(T) Cluster beam

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Melting of Na147 .

++

energy

P(E

)


Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

+4h

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Melting of Na147 .

++

6h5h3h

3h +4h 6h


cou

nts

+4henergy

P(E

)


Photo-disintegration

Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

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Melting of Na147 .

++

3h 4h 5h 6h


cou

nts

3h 4h 5h 6henergy



Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

P(E

)

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Melting of Na147 .

++

3h 4h 5h 6h


cou

nts

3h 4h 5h 6henergy

P(E

)



Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

Tem

per

atu

re


3h 4h 5h 6h


Melting

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Melting of Na147 .

++

3h 4h 5h 6h


cou

nts

3h 4h 5h 6henergy

P(E

)



Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

Tem

per

atu

re


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Melting of Na147 .

++

3h 4h 5h 6h


cou

nts

3h 4h 5h 6henergy

P(E

)



Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

Tem

per

atu

re


Energy distribution

Expected Photo-dissociation

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Melting of Na147 .

++

3h 4h 5h 6h


cou

nts

3h 4h 5h 6henergy

P(E

)



Variable THelium

Heat Bath

E*(T)

E*(T)+ nh

Laser E=nh

Cluster beam

dete

ctor

dete

ctor

Tem

per

atu

re


Energy distribution

Expected Photo-dissociation

Observed

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Tem

per

atur

e,

Pro

bab

ilit

y P

(E

)

Energy

Energy distribution

Photo-dissociation

3h 4h 5h


Haberland-PRL 2001



M

Ising model


INDRA


INDRA-2001


24

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*****Fin bimodalité

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*****Chapitre 3

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- III -Convex S

Back bending T

Negative heat capacity

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Canonical(Average)


Lattice-gas Model

En

erg

y D

istr

ibu

tio

n

1

10

100

0.1

Liquid Gas



Lattice-gas model

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

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Lattice-gas model

Canonical(Average)


Lattice-gas Model

En

erg

y D

istr

ibu

tio

n

1

10

100

0.1

Liquid Gas

E distribution P =W(E)e - E/ Z

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

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Canonical(Average)


Lattice-gas Model

En

erg

y D

istr

ibu

tio

n d

1

10

100

0.1

Liquid Gas

Convex Entropy

Microcanonical

Microcanonic: S=logW log Z P =S(E) - E

Entropy

Negative Heat Capacity

Energy (A MeV)-2 -1 0 1 2 3

3.6

3.5

3.4

3.3

3.2

Tem

per

atu

re (

MeV

)

Decreasing T: T-1= dES

T-1 = - dE(logP

Lattice-gas model

26

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A negative heat capacity

If the energy is fixed (microcanonical)

<=

>

28

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A inverted curvature of the thermodynamical potential

If the observables (order param.) are fixed (intensive ensemble)

<=

>

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First Order Phase Transition in Finite Systems .

Lynden-Bell et al, -on. Not. R. Astr. 138(1968)495, cond-mat/9812172

STARS

30

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Excitation Energy (MeV)

10 2 3 4 5 6

Tem

per

atu

re (

MeV

)

0

0.4

0.8

0.

0.4

0.8

0.

0.4

0.8

1.2172Yb

166Er

162Dy

Mel

by-

1999

Nuclear superfluidity

Gob

et-2

002

Fragmenting H-clusters

F. G

ulm

inel

li, P

h. C

h. P

RL

(199

8)

Canonical liquid-gas

Abnormal curvatures

Convex entropy T decreasing with E

Compressibility < 0Susceptibility < 0

30

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***** bébut superfluid

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At low energy

Melby et al, PRL 83(1999)3150

Decay: level density

X(3He, 3He ’)X*

W(Ef)

32

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Excitation Energy (MeV)10 2 3 4 5 6

Mic

roca

no

nic

al T

(M

eV

)

0

0.4

0.80.

0.4

0.80.

0.4

0.8

1.2

166Er

162Dy

172Yb

Breaking of pairs

Superfluid Nuclei

Melby et al, PRL 83(1999)3150

Decay: level density

Heat

Cap

aci

ty(k

B)

T(MeV)0.5 1.00.0

20.

0.

40.

172Yb

Canonical

T1 logWE

Microcanonical

Superfluid

A.Schiller et al. 2000

X(3He, 3He ’)X*

W =

exp

S

33

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10 2 3 4 5 6

Tem

per

atu

re (

MeV

)

0

0.4

0.8

0.

0.4

0.8

0.

0.4

0.8

1.2172Yb

166Er

162Dy

Mel

by-

1999


Gob

et-2

002


F. G

ulm

inel

li, P

h. C

h. P

RL

(199

8)


Abnormal curvatures



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*******Fin C neg

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Negative heat capacities Abnormal energy fluctuations

Multifragmentation

- IV-

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Microcanonical: fluct. partial energy

Wtot = Wk Wp

E2 = 0, k

2 = p2

2 log Wtot = -1/C f(k

PC, Gulminelli NPA(1999)

Heat capacity from energy fluctuation

Canonical: fluct. EtotZtot = Zk Zp

E2 k

2 + p2

2 log Zi = Ci / i

2

2

1can

kk

C

C

−=

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Microcanonical:fluct. partial energy

Wtot = Wk Wp

E2 = 0, k

2 = p2





E2 k

2 + p2

2 log Zi = Ci / i

2

2

1can

kk

C

C

−=

35

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Microcanonical:fluct. partial energy

Wtot = Wk Wp

E2 = 0, k

2 = p2





E2 k

2 + p2

2 log Zi = Ci / i

2

2

1can

kk

C

C

−=

35

Excitation Energy (AMeV)

MULTICS

MULTICS

D’Agostino et al.

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Sort events in E* (Calorimetry)

Multifragmentation experiment

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Sort events in E* (Calorimetry)


Reconstruct a freeze-out partition

• E2=

iB

i+E

coul

1- Primary fragments: mi

• <E1>=<

ia

i> T2+ 3/2 <M-1> T

3- Kinetic EOS:

• E1=E*- E

2=> <E1>, 1

T, C1 ,2 = C1 T2can

2

2

1can

kk

C

C

−= Deduce C:

Ecoul2- Freeze-out volume:

37

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Large fluctuations

D’Agostino et al.

C12

C1 2/ T2 C =

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Large fluctuations


Ck T2

Ck T2 2

2

D’Agostino et al.

Negative CPeripheral Au+Au eventsC1

2

C1 2/ T2 C =

39

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Flu

ctu

atio

n, C

1



D’Agostino et al.

MULTICS

Hot Au, ISIS collaboration

ISIS

ISIS

MULTICS

Ni+Ni, 32 AMeV Quasi-Projectile


INDRA

Bougault et al Xe+Sn central

0 2 4 6 80

2

1

INDRA

2

2

1can

kk

C

C

−=Multics E1=20.3 E2=6.50.7Isis E1=2.5 E2=7.Indra E2=6.0.5

Excitation Energy (AMeV) Excitation Energy (AMeV)

Flu

ctu

atio

n, C

1H

eat

Cap

acit

y

Hea

t C

apac

ity

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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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***** bimodal fragments 2

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Ni+Au INDRA data O.Lopez 2001

B. Borderie, 2001

Xe+Sn INDRA data

Distribution of events.Largest fragment/second largest

Charge big - small

Bimodality in event distribution Multifragmentation of nuclei

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CO.Lopez et al. 2001, B.Tamain et al. 2003

Z1

Z2 "reservoir" "reservoir"

T

197Au

197AuAu+Au 80 A.MeVINDRA@GSI data

Z1-Z2

Z1

Order parameter Z1-Z2 (big-small)

cf

€

ρL−ρG

Largest fragment/second largest


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***** Chapitre 5

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Conclusion

Negative heat capacitiesBimodality, fluctuations

- V -

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Re

Im

Complex

€

c Order param. free (canonical)

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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 par. free (canonical)

Order par. fixed (microcanonical)

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

€

Pβ0 E( )

E1 E2

E distribution at

Energy

36

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D’A

gost

ino

PL

B 2

000

Boiling nuclei

Tem

per

atur

e, P

roba

bili

ty P

(E

)

Energy distribution

Photo-dissociation

3h 4h 5h


Haberland-PRL 2001

Melting of Na ClusterFragment asymmetry (Zbig-Zsmall)

INDRA


INDRA-2001


10 2 3 4 5 6

Tem

per

atu

re (

MeV

)

0

0.4

0.8

0.

0.4

0.8

0.

0.4

0.8

1.2172Yb

166Er

162Dy

Mel

by-

1999


Gob

et-2

002


Seen in experiments

Energy

4

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noir

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**** Debut V-E lattice-Gas

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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


Isobare ensemble

P(i) exp-V(i)

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Liquid-gas

Phase transitionAnd bimodality

Negative compressibility


Isobare ensemble

P(i) exp-V(i)

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Liquid-gas

Phase transitionAnd bimodality


Order parameter


Isobare ensemble

P(i) exp-V(i)

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Tem

per

atur

e,

Pro

bab

ilit

y P

(E

)

Energy

Energy distribution

Photo-dissociation

3h 4h 5h


Haberland-PRL 2001



M

Ising model


INDRA


INDRA-2001


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noir

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**** Ising début

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Magnetization in Ising Model

One Phase

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One PhaseBimodalTwo Phases

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One PhaseBimodalTwo Phases

Second Order

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5) Ising

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Phase transitiona bifurcation

M

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Tem

per

atur

e,

Pro

bab

ilit

y P

(E

)

Energy

Energy distribution

Photo-dissociation

3h 4h 5h


Haberland-PRL 2001



M

Ising model


INDRA


INDRA-2001


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***** Début bimod frag

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Ni+Au INDRA data O.Lopez 2001

B. Borderie, 2001

Xe+Sn INDRA data

Distribution of events.Largest fragment/second largest

Charge big - small


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Bimodality in event distributionMultifragmentation of nuclei

1 4 C32India, 2006

CO.Lopez et al. 2001, B.Tamain et al. 2003

Z1

Z2 "reservoir" "reservoir"

T

197Au

197AuAu+Au 80 A.MeVINDRA@GSI data

Z1-Z2

Z1

Order parameter Z1-Z2 (big-small)

cf

€

ρL−ρG

Largest fragment/second largest


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C

Tem

per

atur

e,

Pro

bab

ilit

y P

(E

)

Energy

Energy distribution

Photo-dissociation

3h 4h 5h


Haberland-PRL 2001



M

Ising model


INDRA


INDRA-2001


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noir

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***** Début neg comp

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C

Density: Order parameter: Usually V=cst

(“box”) Isochore

ensemble Negative

compressibility

Canonical Lattice-Gas

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

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Generalization: inverted curvatures

Negative Susceptibility and Compressibility



Particle Number

Pressure

Ch

emic

al P

oten

tial

Gulminelli, PC PRL99

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C

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C


10 2 3 4 5 6

Tem

per

atu

re (

MeV

)

0

0.4

0.8

0.

0.4

0.8

0.

0.4

0.8

1.2172Yb

166Er

162Dy

Mel

by-

1999


Gob

et-2

002


F. G

ulm

inel

li, P

h. C

h. P

RL

(199

8)


Abnormal curvatures



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noir

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***** début H clusters

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Fragmentation of hydrogen clusters

Gobet et al, PRL (2002)

H3+(H2)n

He

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C

H2



H3+(H2)n’

He

H2H2

HH

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C



H3+(H2)n’

He H2

H2H2

H

H

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H3+(H2)n’

HeH2

H2H2

H

H

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H3+(H2)n’

HeH2

H2

H2

HH

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H3+(H2)n’

He

H2H2

H2

HH

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Partition => energy Partition => energy

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Partition => energy

Largest => Temperature

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Partition => energy

Largest => Temperature

Cooler

Hotter

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C

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C


10 2 3 4 5 6

Tem

per

atu

re (

MeV

)

0

0.4

0.8

0.

0.4

0.8

0.

0.4

0.8

1.2172Yb

166Er

162Dy

Mel

by-

1999


Gob

et-2

002


F. G

ulm

inel

li, P

h. C

h. P

RL

(199

8)


Abnormal curvatures



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noir

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***** Appendixe flow

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-Appendix - I -Phase transition under flows

Exact theory for radial flow Isobar ensemble required Phase transition not affected

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

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C

€

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”

Equilibrium = Max S under constrains

Additional constrains: radial flow <pr(r)> Additional Lagrange multiplier (r)

Self similar expansion <pr(r)> = m r => (r) = r

Does not affect thermo is flow subtracted 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∑( )

Thermal distribution in the moving frame

Negative pressure Isobar ensemble

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Radial flow at “equilibrium”

Does not affect thermo is flow subtracted Gulminelli, PC, Nucl-Th (2002)

xz

Expansion

Does no affect r partitioning

Only reduces the pressure

Shifts p distribution Changes fragment

distribution: fragmentation less effective

pres

sure

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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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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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Affects p space and fragments

Gulminelli, PC, Nucl-Th (2002)

px

pzTransparency

Fragmentation less effective If Eflow not subtracted

Affect <E> Affect 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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***** Appendix Coulomb

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- Appendix -II-Thermo with Coulomb

Coulomb reduces coexistence and C<0 region

Statistical treatment of Coulomb (EN,VC) a common phase diagram for

charged and uncharged systems

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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( )

C Uncharged C N Charged

€

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

C N Charged

C Uncharged

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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

C N Charged

C Uncharged

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Energy

12

34

Cou

lom

b in

tera

ctio

n V

C

0 2 4 6 8 10 12 14

C N Charged

C Uncharged

Partitions may differ for heavy nuclei Channels are

different (cf fission)

Re-weighting impossible

However, a unique phase diagram

(EN VC)


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noir

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***** Appendix equilibrium

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-Appendix - III -Equilibrium in finite systems

Macroscopic One realization (event) at equilibrium

Microscopic Statistical ensemble and information Ergodicity versus mixing dynamics Multiplicity of equilibria What is temperature?

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Equilibrium : What Does It Mean? Finite systems

In time In space

Isolated open systems No reservoir or bath No container

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Equilibrium : What Does It Mean?

Thermodynamics : infinite system

One ∞ system = ensemble of ∞ sub-systems

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Finite system

Cannot be cut in sub-systems

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if it is ergodic


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


One small system in time

Many events

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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>

CanonicalIsochoreIsobareDeformedExpandingGrandRotating

... Others

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What is temperature ?T

W equiprobable microstates

€

S=klogW

L. Boltzmann

Entropy = disorder (-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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It is what thermometers measure.

E = Ethermometer + Esystem


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E = Ethermometer + Esystem


Distribution of microstates

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S = k logW


The microcanonical temperature


Distribution of microstates

S =T-1 E

Eth

E

P(E

th)

0 0

E = Eth + Esys

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Most probable partition: Tth = sys


The microcanonical temperature


Equiprobable microstates

max P => logWth - logWsys =0

S =T-1 E

Eth

E

P(E

th)

0 0

P(Eth) Wth (Eth)* Wsys(E-Eth)

E = Eth + Esys

S = k logW

max P => Sth - Ssys =0

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noir

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****** Appendix volume

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-Appendix -IV-C<0 in Liquid gas transition

Negative heat capacity in fluctuating volume ensemble

Difference between CP and CV

Difference between C and E(T) Role of the thermo transformation

Negative heat capacity in statistical models and Channel opening

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- a -Role of volume fluctuationsFixed volume Cv>0 but K<0 Open systems Cp<0

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

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

6

8 T=10 MeV


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


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.

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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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- b -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


2-D EOS

Energy

Tem

per

atu

re

Pressu

re

E TV P

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Lattice-Gas Model


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


Energy

Tem

per

atu

re

Pressu

re


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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- c -Negative curvaturesChannel opening Ex: Statistical models

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C<0 in statistical modelsLiquid-gas and channel opening

q = 0 =>

no-Coulomb

Many channel opening

One Liquid-Gas transition (Vc related to V) 0.

40.

81.

21.

62

Cou

lom

b in

tera

ctio

n V

C

0 2 4 6 8 10 12 14Nuclear energy

Pro

bab

ilit

y

Probability

€

H =HN+q2VCBimodal P(EN)

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noir

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***** Appendix iteration et correlations

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-Appendix -V-How to progress

Correcting errors in reconstruction

Iterative procedure

C from other observables Using correlated observables

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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) k can be corrected

Iterate the procedure Freeze-out reconstruction Decay toward detector

Reconstructed freeze-out fluctuation

Corrected freeze-out fluctuation

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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

−=

k k correlation “canonical” fluctuations

Ek

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noir

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***** Appendix test hypothèses

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-Appendix -VI-Test of freeze-out reconstruction

Volume Coulomb boost

Temperature Isotope ratios Particle-fragment correlations

Alternative extraction

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Sort events in energy (Calorimetry)


• 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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Sort events in energy (Calorimetry)


• 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

E*

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C

Volume

From Coulomb boost C Weakly sensitive to V

Au QP

6V0

3V0

M.D

’A

gosti

no N

PA

2002

<E

k>

Eflow (A.MeV)

3 4 5 6V0

exp.

central Xe+Sn

R.B

ou

gau

lt 2

002

Ambiguity with expansion

Tiny influence on C

Need isotopic resolution

mass number

<E

k> Z=cte

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CM.D’Agostino et al. 2001

3V0

6V0

a=12 a=8

C Ck/(Ck-k/T2)

Ek = E*-Q-Vc(V)

<Ek> = aiT2 + 3/2(<m>-1)T

Ck = d <Ek>/dT Ck<1.5 Atot

2 2Au+Au 35 A.MeV

MULTICS data

Influence on C

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C

T C isotope

M.D ’Agostino NPA2002

Evaporation and temperature

<Ek> = ai(T)T2 + 3/2(<m>-1)T

Consistency of T

Correlation

Au QPS.Hudan 2002

A/8

A/13

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Heat capacity from the translational energy

( )( )2

2 2)2/3(1)2/3(1

1

1)2/3(

tr

tr

E

mmT

C

m

ET

−−−=

−=

int* EVQEE ctr −−−=

calorimetry

Coulomb trajectories

LCP-IMF correlations

NPA699(02)795

Multics Au+Au 35A.MeV

E.M.Pearson 1985, A.Raduta 2002

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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

gosti

noIW

M2

00

1

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noir

Phase transition in finite systems

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