1. Fluctuations of fragment observables
A review of the theoretical and experimental status of the art
M.D’Agostino F.Gulminelli
Introduction: motivation
finite size effects
Theory: fluctuations and constraints
fluctuations and susceptibilities
Experiment: M2, g2, NVZ, sk
Open questions: quantitative estimation of fluctuations
an ideal gas of fragments ?
the dynamics of the expansion

2. Introduction

3. Historical motivation
Introduction I
Historical motivation
Diverging fluctuations at a critical point
Multifragmentation: opening of phase space
large fluctuations of fragment partitions
Nautilus 93
Aladin 96
Isis 01

4. Caveat: finite size effects
Introduction II
Caveat: finite size effects
3D percolation
(1) Smoothing of the phase transition
(2) Conservation constraints and sorting effects
X.Campi 88
3D percolation
3D percolation
random partitions
J.Elliott 00

5. (3) Thermodynamic ambiguities :
Introduction III
(3) Thermodynamic ambiguities :
independence of the density of
the fluctuation behavior
J.Pan 98
J.Carmona 98
P.Chomaz 00

6. Need to compare with models and/or quantify the fluctuation peak

7. Theory

8. Fluctuations and constraints
Theory I
Fluctuations and constraints
c.e.
V.V.Begun 05
grancanonical
canonical
microcanonical
Ntot=N++N- N+
Mass conservation
versus
fluctuations of Amax
J.Carmona 03
F.Gulminelli 03
Fluctuations are suppressed if a
constraint is applied to a correlated variable

9. Fluctuations and constraints
Theory II
Fluctuations and constraints
F.Gulminelli 01
Volume constraint
versus
energy/Amax fluctuations
J.Carmona 03
Fluctuations are suppressed if a
constraint is applied to a correlated variable

10. Relation between fluctuations and susceptibilities
Theory III
Relation between fluctuations and susceptibilities
P.Chomaz 01
Conservation constraint
Statistical independence
Gaussian approximation
from fluct.
exact
where
and sref variance of A1
in the ensemble where only
<A> is constrained
Ex:

11. Data

12. Effect of the sorting variable
Experiment I
Aladin 96 20 40 60
Zbound
AGS-BNL 03
Au+Au
400,600
800,1000
A.MeV
Au+emulsions
Au+Au
35 AMeV
Aladin 96
AGS–BNL 94
Multics 00
Au+C
1AGeV
EOS 00

13. Size effects Experiment II g2 E(max) ~ 5 A.MeV EOS 03 Multics 00
NVZ=sNV=s2Zm/<Zm>
RMS= sZm/<Z0>
Size effects
Experiment II
Au+C
RMS~.15
Nimrod 04 g2
Ar+Al,Ti,Ni
La+C
Au+Au
Multics 00
Kr+C
Nimrod 04
Ar+Al,Ti,Ni
RMS~0.03
<E*/A>(MeV)
Multics
H clusters
s/sc
E(max) ~ 5 A.MeV
m/mc

14. « Configurational energy » fluctuations
Experiment III
« Configurational energy » fluctuations
Multics 04
average kinetic energies
lcp-fragment correlations
+ hypothesis on mass

15. « Configurational energy » fluctuations
Experiment IV
« Configurational energy » fluctuations
Indra
Ni+Au 50AMeV
Indra
Xe+Sn 32AMeV
Multics-Miniball
Au+Au 35AMeV
Au+C 35AMeV
Au+Cu 25,35AMeV
Indra+Aladin
Multics 04
Pre-equilibrium?
Radial flow?
Neck emission?
Au+Au 80AMeV

16. « Configurational energy » fluctuations
Experiment V
« Configurational energy » fluctuations
Multics 04
Nimrod 04
Size effect?
Sorting?
Side feeding?

17. « Configurational energy » fluctuations
Experiment VI
« Configurational energy » fluctuations
Multics 00
As +- 10%
no restriction
Au+C
1AGeV
Au+Au 35A.MeV
Kr+C
1AGeV
Source size conservation?
Flow effects?
Are we looking at the same observable?
E*/A
E*/A
EOS 02

18. Open questions

19. From theory to data: an ideal gas of fragments ?
Open questions I
From theory to data: an ideal gas of fragments ? ?
Campi Krivine 04
Lennard Jones CMD
thermo
QLD
aV(l),aS(l)
Gulminelli Chomaz 04
Lattice Gas EI
QLD

20. Out of equilibrium effects: the dynamics of the expansion
Open questions II
Out of equilibrium effects: the dynamics of the expansion
How far is a freeze out from thermal
equilibrium?
E=-1.7e
E=0.1e
E=0.8e
asymptotic
t=0 confined
Chernomoretz 04
time
t=0 confined
asymptotic
Balenzuela 02

21. Conclusions
The theoretical connections between fluctuations and susceptibilities are well established but conditions have to be checked
conservation constraint
statistical independence
gaussian distribution
Experimental multiplicities/sizes fluctuations agree reasonably well (in comparable size ranges)
The methodology to extract configurational energy fluctuations is under debate
cluster definition in dense quantum media (Fermi motion?)
the dynamics of the expansion + entrance channel
systematic errors in fluctuations measurements
The challenge is finite systems thermodynamics: it is worthwhile to go on
We need – full comparisons with a well defined protocol
and consistency checks
- better measurements to assess the effect of the experimental constraints

22. Out of equilibrium effects: uniformity of phase space
Open questions VI
Out of equilibrium effects: uniformity of phase space
Gulminelli Chomaz 02
A.Majka 03
M.Ploszacjack 02
Tsallis entropy