1. WCI3 Town Meeting, College Station (Texas)
Bimodalities
A compilation of models and data
O. Lopez, M.F. Rivet
Good afternoon, ladies and gentlemen!
First of all, I would like to thank the organizers for the invitation to come to this WCI meeting.
In this presentation, we will report on bimodalities from both experimental and theoretical sides. At the end of the talk, Marie-France and myself will lead the discussion of 30 minutes about the issues concerning bimodalities.
WCI3 Town Meeting, College Station (Texas)
February,

2. Bimodalities part 1 Theoretical bases
First we will recall the definition of bimodality in Thermodynamics.
LOPEZ/RIVET, WCI3 Town Meeting, College Station (Texas) February,

3. Bimodalities : foundation
Bimodality of the magnetization
in a finite Ising ferromagnet at
phase transition
Binder & Landau PRB30 (1984)
Bimodalities and earlier studies
Bimodalities and its relationship to phase transition has been studied since the eighties as presented here, where we can see a Ising model simulation of a ferromagnet. In this study, the authors study the magnetization as a function of the applied magnetic field H. When the magnetic field is close to the critical value Hc, the ferromagnet presents a jump in its magnetization; this jump is a step function for an infinite system as indicated by the red curve. It is worthwhile to mention that in this case, no bimodality in the magnetization can occur, the system jumps suddenly from the negative spontaneous value -Msp to its positive value +Msp.
In Finite systems
By contrast, when the size of the system is finite (L), the step function is replaced by a rounding curve (in blue) with a slope which goes as Ld (d is the dimensionality). When we look in the vicinity of Hc, we can then see a bimodal behavior for the magnetization as shown by the bottom figures.
In brief, the bimodality in the order parameter of the system (here M) is present only for a finite size L, and is then characteristic of a finite system.
Characteristic of a finite system
slope at Hc  Ld/T
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4. Bimodality of order parameter distribution
new definition of a 1st order phase transition in finite system
Chomaz et al. PRE64 (2001)
1/ Equivalent to Yang-Lee theorem (stand. def. at thermodynamic limit)
Chomaz/Gulminelli Physica A330 (2003)
and to
In standard thermodynamics, the bimodality of order parameter is related to a 1st order phase transition of a system. In this slide, we can see 3 different but equivalent definitions of phase transition and the relationship between it and bimodality.
Yang-Lee theorem
It has been shown by Chomaz and Gulminelli that bimodalities of order parameter is equivalent to the standard definition of phase transition in the thermodynamic limit when we recover the usual definition by the zeroes of the partition function in the complex temperature plane.
Anomaly of thermodynamical potentials
A first order phase transition is characterized by an inverted curvature of the relevant thermodynamical potential (entropy, free energy and so on). This feature is also equivalent to a bimodality in the event probability of the order parameter X.
Negative derivatives of the thermodynamical potentials
This is also related to a back-bending in the equation of state of the system, characterized by a negative heat capacity in case of Energy as order parameter.
2/ Inverted curvature of thermodynamical potential
Chomaz, Colonna, Randrup, Phys. Rep. 389 (2004)
3/ If X=E*, negative heat capacity
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5. Phase transition definition and statistical ensembles
C<0 : the order parameter (E* ) must be controlled
microcanonical ensemble
Bimodality and Yang-Lee Th.: extensive variable (E*, V )
should be free to fluctuate
Thermodynamical ensembles
Shortly, we recall here that for studying the abnormal energy fluctuations (negative heat capacity), we need to control the order parameter of the system, here the excitation energy, whereas in case of bimodality, no condition upon energy or any order parameter should appear; by contrast, the energy should be free to fluctuate in order to fill correctly the corresponding accessible phase space. The correct ensemble is then the canonical ensemble, where only the temperature T is controlled in average.
Canonical ensemble
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6. Relaxed condition: Gaussian ensemble as interpolating ensemble
N particles in sample, N’ in heat bath, vary N’ from 0 to ∞
a  1/N’
Bimodality obtained
for large N’ (small a)
≈ canonical
and not for small N’
≈ μcanonical
Canonical vs. microcanonical
Illustration of the previous statements
Gaussian ensemble
The figure presented here is the result of a simulation where N particles of a subsystem are in contact (heat bath) with N’ particles at temperature T. Hence, when N’ goes to infinity, we are in a canonical situation while when N’=0, the system is microcanonical. This is called the Gaussian ensemble.
Appearance of bimodality
In this simulation, we see clearly that bimodality in the energy distribution of the subsystem of N particles appears only when the ratio N/N’ is of the order of 1/1000 at least; for greater N/N’, the situation is close to the microcanonical case with no bimodality.
Challa & Hetherington PRL60 (1988)
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7. LG Phase transitions : different possible order parameters
Canonical Lattice gas
model in the first order
phase transition region.
Chomaz et al. PRE 64 (2001)
V (rliq-rgas)
Energy E
any linear comb. of E,V
Order parameters are :
Liquid-Gas phase transition
Now, let’s go to the specific case expected in the nuclear case : the liquid-gas phase transition. We show here results coming from a Lattice-Gas model where the LG phase transition is present.
Order parameters
When we are in the canonical framework and look at the event probability distribution in the first order phase transition region, we see a nice bimodal distribution in the Volume vs. Energy plane (top left). The projections along the axes (E,V) show a bimodality as expected, and again when looking at a linear combination of these 2 order parameters (bottom right, red curve).
Constraints on data analysis
In this case, the bimodality can be seen if we are able to select (sort) events in a canonical way (or as close as possible), and plot the event probability distribution of the energy or volume, or at least observables directly related to them.
We will now look at the experimental observations of such a signal.
LOPEZ/RIVET, WCI3 Town Meeting, College Station (Texas) February,

8. Bimodalities part 2 Experimental facts
Now, you will look at the experimental data, and first we apologize if we have forgot a result or a study concerning bimodalities. This presentation has been made with the cooperation of people which have provided us materials before the meeting.
LOPEZ/RIVET, WCI3 Town Meeting, College Station (Texas) February,

9. Bimodalities: experimental overview
52 MeV/nucleon
Zone 2
Zone 1
E*/A (MeV) (calorimetry) DZ
E*/A ≠ (1)
Bellaize et al,
Nucl. Phys A709 (2002)
INDRA collaboration
Bimodalities: experimental overview
32/52/90 MeV/nucleon 58Ni+197Au
A ~ , Central collisions (PCA)
Observation of a
bimodality in fragment size asymmetry
for E*~4 (32) and E*~5 (52) MeV/n (from SMM)
Central collisions, from heavy to light systems
First of all, we will look at experimental data for central collisions where we are in the best situation to see the phase coexistence and phase transition.
Ni+Au INDRA data
3 years ago, Bellaize and myself have reported the observation of a bimodality in the size asymmetry of the 3 largest fragment in a central event (the system was Ni+Au at and 90 MeV/nucleon) recorded with INDRA. This bimodality is associated to 2 fragmentation patterns, one (called here Zone 1) corresponding to residue-evaporation (a large fragment with no IMF or very few) , the other to multifragmentation (fragments of nearly equal size, zone 2).
This feature is clearly seen when we look at the differences between the charges of the 3 largest fragments as shown in the bottom row; at 32 and 52 MeV/nucleon, a bimodality is present while it has disappeared at 90 MeV/nucleon.
Different excitation energy
If we select the 2 patterns and look at the excitation energy (experimentally deduced from the energy balance), we can see that the excitation energies are slightly different (actually they differ from 1 MeV/nucleon), as expected if the order parameters are the ones of the LG phase transition.
E* differs from 1 MeV/nucleon
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10. SMF simulation (BOB) Xe+Sn 32A MeV
Bimodalities: experimental overview
Borderie et al,
Journal of Physics G (2002) R217
INDRA collaboration
32-50 MeV/nucleon 129Xe+natSn
A ~ , Central collisions (qflow)
SMF simulation (BOB)
Xe+Sn 32A MeV
Central collisions Xe+Sn MeV/nucleon, INDRA data
Borderie et al have also reported the observation of bimodality, but this time by looking at the fraction between Z greater than 12 and Z comprised between 3 and 12. When looking at the different bombarding energy (32 to 50 MeV/nucleon) in the right panel, we see the two contributions which evolve as expected; the symmetric channel (left peak) is raising as a function of the incident energy while the asymmetric one (right peak) is decreasing accordingly. The crossover between the 2 patterns is situated around 39 MeV/nucleon (red triangles).
Stochastic Mean-Field simulations
To get more information, a stochastic mean-field simulation (BOB) has been done and compared to the data; it is in good agreement as reported in the paper and the size asymmetry fraction is plotted in the left panel for the simulation at 32 MeV/nucleon; the distribution resembles to the one of data (black stars) and allow to conclude that the scenario of the crossover of the system through the coexistence region (where spinodal instability can set in) is probable.
Ratio between “Liquid” and “Gas” phase
Passage through the coexistence region
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11. Bimodalities: experimental overview
Lautesse et al,
PRC 2005, in press
INDRA collaboration
32 MeV/nucleon 58Ni+natNi
SMM
GEMINI
A ~ 100, central collisions (DFA)
(Z1-<Z>)2+ (Z1-<Z>)2+ (Z2-<Z>)2
Asym123 =
6<Z>
Central collisions, Ni+Ni at 32 MeV/nucleon, INDRA data
In this study by Lautesse and co-workers, they analyzed central collisions sorted with a Discriminate Factorial Analysis. They found that when looking at the asymmetry variable build with the 3 largest fragment in an event in a way indicated here, they also obtained 2 fragmentation patterns; a more detailed analysis showed that the 2 components are compatible respectively with GEMINI (sequential) and SMM (simultaneous) as presented on the figure. These results are then compatible with a bimodality in the size asymmetry in fragments produced in these collisions corresponding to a source size of 100.
Data exhibit 2 fragmentation patterns:
Evaporation-Residue (GEMINI-like)
multifragmentation (SMM-like)
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12. Bimodalities: experimental overview
M. Pichon, Thèse LPC Caen (2004)
80 MeV/nucleon 197Au + 197Au
Etrans12QT bins
“canonical” sorting
A = , peripheral reactions (QP)
E* ≠
Zasym<0.3
Zasym>0.7
Equal T
Peripheral reactions, Au+Au at 80 MeV/nucleon, INDRA data.
Now let’s look at peripheral reactions. Let’s see if we observe that kind of signal. As already mentioned, the events have to be selected canonically. How doing this for peripheral reactions?
“Canonical” sorting
In a recent analysis, Pichon and co-workers have used the transverse energy of quasi-Target particles to mimic a canonical sorting, ensuring that the selection allows to study without strong and trivial autocorrelations the Quasi-Projectile source.
The results
The results presented in this panel show the correlation between the largest fragment in the QP source and the asymmetry variable defined as the normalized difference between the largest and the second largest fragment in the QP source. The different panels correspond to the transverse energy selection. We see clearly that in panel 3 (top right), two components are present at the same time.
Liquid-gas phase transition?
By analyzing further this 2 sets of event, we can see that they correspond to different excitation energies (in green for the large asymmetries and in blue for the small ones). At variance, if we look to the measured temperatures, we find that they are identical, supporting then the fact this bimodality is compatible with what one could expect from a liquid-gas phase transition.
Compatibility with the LG phase transition
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13. QT light particles transverse energy
Bimodalities: experimental overview
D'Agostino et al,
5th Italy-Japan Symposium – Naples 2004
MULTICS-MINIBALL collaboration
35 MeV/nucleon 197Au+197Au
A ~180, peripheral reactions (QP)
QT light particles transverse energy
Peripheral collisions, Au+Au at 35 MeV/nucleon, MULTICS/MINIBALL.
Here, we see results collected by the MULTICS/MINIBALL collaboration; they analyzed in the same way the fragment asymmetry in peripheral collisions coming from the Quasi-Projectile source; they sorted in the same manner the events and found similar results, but for lower incident energy here, at 35 MeV/nucleon.
The same signal is observed
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14. Interpreted as a 2nd order PT ?
Trautmann et al,
private communication (2005)
ALADIN collaboration
Bimodalities: experimental overview
1000 MeV/nucleon 197Au+197Au
Zbound
Zb3
Zbound=53-55
A ~130, peripheral collisions (QP)
Same behavior
Interpreted as a 2nd order PT ?
Bond percolation (53), pb=0.328
Peripheral collisions (QP), Au+Au at 1 GeV/nucleon, ALADIN data.
Trautmann and co-workers looked to the data they have collected within the ALADIN collaboration and found that they were able to observe a bimodal behavior in the charge asymmetry defined as the difference between the 3 largest fragments of the QP source. In this figure, we see the Zbound selection (53-55), and the corresponding charge asymmetry.
Percolation simulation
They realized a bond-percolation simulation, as mentioned in this reference, and when they looked around the critical bond probability (here 0.328), they observed the same feature for the charge asymmetry and conclude that they observe here a phase transition.
A contradiction?
Nevertheless, the percolation is a model where one can observe second order phase transition and not first order ! The issue of the order of the phase transition is thus not so clear, and I will come back to this point by the end of the talk.
Wait final remarks …
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15. Bimodalities: experimental overview
Ma et al,
Nucl-ex/
NIMROD Collaboration
47MeV/nucleon 40Ar+27Al,48Ti,58Ni
A = 24-40, peripheral reactions (QP)
Excitation energy is deduced from
3-sources fits and calorimetry
Peripheral collisions, Ar+Al/Ti/Ni at 47 MeV/nucleon, NIMROD data.
In this very complete study, Ma and co-workers analyzed data collected with the NIMROD array. They studied QP particles and fragments coming from the quasi-Argon in peripheral and semi-peripheral collisions.
Excitation energy
They first deduced excitation energy by tagging the particles with the help of 3 moving sources fits and then attribute a event probability for a particle to be emitted by one of these sources (QP, QC and mid-rapidity). They then applied the energy balance and obtained the excitation energy distributions shown in the left panel for the 3 targets. The distributions collapse all together showing that the QP excitation energy calculation is under control.
Bimodality
They have then looked to an observable they called the bimodal parameter which is defined in similar way as for the Borderie’s analysis by the normalized difference between the sum of Z greater than 2 (“liquid phase”) and the sum of Z less or equal to 2 (“gas phase”). They found that the equilibrium between the two phases is obtained for E*/A=5.5 MeV, which is the value where they observed the largest fluctuations of the largest fragment, the minimum power-law exponent for the charge distribution, the scaling laws, and so on…
A crossover between “gas” and “liquid” phase for E*/A ~ 5.5 MeV (exc5-6)
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16. Bimodalities part 3 Models and simulations
Enough for the experimental data, now we will look at models prediction concerning bimodalities.
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17. Bimodalities: Statistical Multifragmentation Model
nucl-th/
N. Buyukcizmeci, R. Ogul, A.S. Botvina
Calculations for different nuclei
From E*A=2-20 MeV
2 classes of events are seen
depending on the size of Amax
(A)
- Amax>2A0/3 : residue-evaporation
- Amax< A0/3 : multifragmentation
First of all, the Statistical Multifragmentation model (SMM)
Buyukcizmeci, Ogul and Botvina analyzed SMM simulations of different nuclei, for excitation energy varying from 2 to 20 MeV/nucleon. They found that the nuclei gave the same caloric curve as depicted in the top panel, with the well-known “plateau” between 4 and 8 MeV/nucleon.
Sorting with Amax
They have then sort the events following the size of the largest fragment (Amax). They defined 2 event classes, one with Amax greater than 2 third of the total system size, and the other one with Amax less than 1 third. The third panel show the corresponding probability of the 2 classes and we observe that a transition occurs around 3-4 MeV/nucleon, which corresponds to the maximal fluctuations in the mass distribution (second panel).
Explanation of the plateau
When they plotted the corresponding caloric curves for the 2 event classes, they observed that the residue-evaporation class is of Fermi-gas type (proportional to T squared) while the prompt multifragmentation class is associated to a classical gas type (linear in T). It is then the combination of these 2 behaviors which gives rise to the plateau zone in the total caloric curve and explained the inflexion point of this curve.
They correspond to different caloric
curves (liquid and gas type)
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18. Bimodalities: Classical Molecular Dynamics
D. Cussol
Private communication (2005)
Bimodality in A1-A2 is observed
for 3 simulation sets :
- central collisions (b<0.1)
- peripheral collisions (QP)
- thermalized systems (r0/r=8)
The transition is located
at different excitation energies
Eleastbound < E*/N < 2Eleastbound
Classical Molecular Dynamics.
A second class of models can be used to look at bimodalities and more generally signals of phase transition, namely dynamical models. The first one is a classical molecular dynamics with a Lennard-Jones potential done by Cussol at LPC Caen. Symmetric collisions of LJ droplets with size of and are analyzed.
Different experimental conditions
In this study, Cussol have looked to a system prepared in 3 different conditions; central collisions (small impact parameters), peripheral collisions (all impact parameters but looking at the forward zone), and “thermalized” systems (particles are placed in a box of volume V/V0=8 during a time sufficient to obtain equilibrium and then released). He looked at the size asymmetry between the 2 largest fragments in an event and here are the results he obtained.
Transition energy
He observed for all conditions the bimodality (the occurrence of both fragmentation pattern), at different excitation energy expressed as the ratio between the excitation energy per particle and the binding energy of the least bound particle. The transition occurs between 1 (central), 1.5 (peripheral) and almost 2 (thermalized).
What is its meaning ?
Role of the deposited energy ?
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19. Bimodality: Heavy Ion Phase Space Exploration
Van Lauwe et al, PRC 69, (2004)
Lopez and Lacroix,
Preliminary results (2005)
~ hbar
20-30 hbar
80 MeV/nucleon Xe+Sn, Et12QC bins (QP)
HIPSE simulation, Xe+Sn at 80 MeV/nucleon.
In this last dynamical model where there is a full (classical) treatment of the entrance channel (nucleus-nucleus potential, NN collisions), followed by a random sampling of nucleon on a Thomas-Fermi distribution to take into account the Fermi motion of nucleons inside nuclei for the participant zone, and then a statistical de-excitation of the formed fragments, including QP and QT if they are still there.
Observation of bimodality in fragment size asymmetry.
The same analysis as for the Pichon’s analysis has been done by myself and D. Lacroix; the same bimodal structure is observed in the plots of Zmax as a function of the charge asymmetry. By looking further, we found that the bimodality is due to the spin transferred to the QP source for semi-peripheral reactions (impact parameter between 4 and 6 fm). For this window of impact parameter a spin value of hbar has been transferred; indeed, it corresponds to the critical value which cancels the fission barrier for such a nucleus (quasi Xenon) and opens the emission of clusters up to symmetric fission as displayed in the panel.
Excitation energy?
In this figure, the charge asymmetry for different excitation energy/spin has been computed for a Xenon nucleus with the help of the SIMON statistical code. If we look at the excitation energy effect, the asymmetry becomes smaller and smaller but never reaches the small asymmetry values; it is only by varying the spin that we obtain this result, and it is very sudden (located between hbar for the Xenon nucleus).
We conclude at that point that we do observe a phase transition, but not of liquid-gas type. This is here governed by the transferred linear momentum into intrinsic spin.
Bimodality and Phase Transition (not LG)
The transition is governed by
the transferred spin to the QP source
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20. Bimodalities part 4 Questions and (some) answers
We have done with the models, now we will finish the talk by asking some important issues and see if we can already give clear answers.
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21. Bimodalities: general overview of experimental data
Results from
Reaction type
Size (A)
Bimodality in
INDRA
Central
~ 200
Z1-Z2
Zliq-Zgas (**)
~ 100
Asym123
Peripheral
MULTICS/MINIBALL
~180
ALADIN
~130
NIMROD
24-40
Zliq-Zgas (*)
Here is a compilation of the experimental data involved in the search of the bimodalities signal. As we can see, the bimodality is always observed in charge asymmetry or related variables, but never in Amax directly.
Can we understand this ?
(*) Zliq= sum of Z>2, Zgas= sum of Z<=2
(**) Zliq= sum of Z>12, Zgas= sum of Z<=12
Amax: a better order parameter ?
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22. In experiments, we never observe a bimodality of Amax,
but for some asymmetry parameter (Aasym, Z1-Z2, …) ,
Why ?
Question
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23. Heat Bath No Heat Bath Bimodality is observed in Aasym but not in Amax
Event sorting is not enough canonical ?
Lattice-gas
F. Gulminelli
Heat Bath
β = βtr
= (SA30- SA2-30)/As
No Heat Bath
E  Etr
A possible explanation
We have seen that to observe bimodality, we have to do a canonical sorting of the data. What happen when it is not the case (as it is certainly in real experiments!) ?
An answer with the Lattice-Gas model
If we look at the LG simulation in the canonical ensemble, we obtain the bimodality for Amax as well as for the charge asymmetry. Fine !
Now, if we do the same simulation but in the microcanonical ensemble, we can see that the bimodality for Amax has disappeared while it is still there for the charge asymmetry.
In other words, the charge asymmetry is a more robust signal than the largest fragment, but we will see that there is a price to pay !
Bimodality is observed in Aasym but not in Amax
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24. What is the influence of out-of-equilibrium effects ?
Question
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25. Bimodality: Lattice-gas model and out-of-equilibrium effects
F. Gulminelli
Phase coexistence in Nuclei
Habilitation Université de Caen (2003)
Bimodality and radial flow
Bimodality is still present
even for 100% of radial flow
(as compared to E)
Similar conclusions are observed
in presence of transparency
(longitudinal flow)
F. Gulminelli and Ph. Chomaz
Nucl. Phys. A734 (2004) 581
What is the sensitivity of the bimodality concerning out-of-equilibrium effects such as radial flow or transparency ?
In the left panel, we see a (canonical) Lattice-Gas simulation taking different prescription for the radial flow energy from 0 to 100% (compared to the thermal energy). If we focus on the top right panel, at the critical temperature T=0.68e, the bimodality in Amax is always observed, even in case of 100% flow.
Similar conclusions are drawn in presence of longitudinal flow (mimic of the transparency).
These 2 examples show the robustness of the bimodality to external (and realistic) constraints due to the dynamics of the collision.
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26. What is the influence of Coulomb interaction ?
Question
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27. (Isobar Microcanonical Multifragmentation Model)
Bimodalities: Influence of Coulomb
Gulminelli et al
PRL 91, Number 20 (2003)
Bimodality in IMMM
(Isobar Microcanonical Multifragmentation Model)
A multicanonical approach allows the mapping between Coulomb and uncharged systems
Inclusion of Coulomb deforms the event probability structure of the phase space but the bimodal character still remains for small systems but not for the heavier ones …
In most models, the inclusion of long range forces as Coulomb is neglected. Nevertheless, since we want to study nuclei which are charged, this issue has to be addressed.
IMMM
In this study, the authors have used a microcanonical multifragmentation model with an constrained intensive variable (here the Pressure, so the volume is conserved in average). They have then compared in the framework of this model the differences between the uncharged system (color contours) and the charged one (lines).
Deformation of the phase space
When there is no Coulomb included, they obtained a nice bimodality in the Vc-Energy plane as expected (at a given temperature). The inclusion of Coulomb deforms the correlation and collapses in each other a little bit the 2 distributions. Moreover, if one is able to estimate properly Vc, it is then possible to recover the “uncharged” bimodality and then see very easily the signal.
The situation is however not so easy for an heavier system as indicated in the bottom panel (Z=82). We see that the effect of Coulomb is not only to deform the phase space but even blur the bimodality signal. So, in the framework of this model, special care has to be done when analyzing size dependence and its relationship to bimodality.
Size dependence ?
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28. Is the asymmetry parameter a good order parameter ?
Last question
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29. Aasy = (Amax –A2)/(Amax +A2)
Bimodality of Amax (order parameter):
only when there is a 1st order PT
No PT
2nd PT
1st PT
F. Gulminelli, private communication
LOPEZ/RIVET, WCI3 Town Meeting, College Station (Texas) February,
Aasy = (Amax –A2)/(Amax +A2)
may lead to
ambiguity on the
order of the PT
but not on the
existence of a PT
In the first proposed question, we have seen that the charge asymmetry is more robust than the largest fragment Amax if we relax the canonical sorting. I have said that there is a price to pay…
No phase transition, no signal.
In the different simulations, random partitions at left, bond-percolation in the middle and lattice-gas at the right, Francesca Gulminelli has checked the response of Amax and the size asymmetry Aasym
First of all, in absence of phase transition (left), none of the observables are bimodal, which is nice !
Order of the phase transition
Now, in presence of a phase transition (2nd or 1st order), we see that the mass asymmetry presents a bimodality while the Amax is only bimodal in the first order case. It shows that for evaluating the order of the phase transition, we have to look in priority at the Amax distribution and not to the mass asymmetry which can be insensitive to it !

30. Bimodalities : some questions
What is the best order parameter: Amax vs. (Amax-A2), (Amax-A2-A3), … ?
What about the event sorting: canonical vs. X ?
Some alternative explanations may exist :
spin, geometry effect, boundary conditions, …
Open questions …
Limitations:
Coulomb, …
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31. Bimodalities : to go further from the exp. side
Correlation between bimodality and the other proposed signals for the LG phase
transition:
Abnormal energy fluctuations (negative heat capacity)
Scaling laws (universal fluctuations, Fisher’s, Zipf’s, …)
Space-time correlations (emission times, correlation functions)
Improve the validity of the signal
Some conclusions from our own …. It should be expanded during the discussion !
Peripheral (QP/QT) vs. central reactions
Influence of the size of the system (varying masses and energies)
Disentangle the entrance channel effects
A+A vs n+A/A+n reactions
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32. BIMODALITIES The discussion is now OPEN Let’s go to it !
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33. The end

35. Bimodalities : summary
Definition:
Bimodality is the presence of two components in an event probability distribution of an order parameter for a finite system; it is related to the coexistence of two phases for the considered system
Generic features:
It is related to the abnormal convexity of the entropy or other thermodynamical
potentials
It is then a 1st order phase transition
A summary about theoretical expectations concerning bimodalities
It should be observed in a “canonical” treatment of data or at least in the framework of “gaussian” ensembles
It is a signature of the liquid-gas phase transition in nuclear matter if
Order parameter is the density (rliq-rgas) or Energy
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36. Bimodalities : what could be in the Consensus
From the theoretical side :
The relationship between phase coexistence and bimodality is clearly established for finite systems
The order parameter for the Liquid-Gas phase transition is (rliq-rgas) or E and is
related to the biggest fragment (Amax) for LG (but not only !)
From the experiment side :
A proposition for the Consensus …
Two decay modes are observed; they correspond to the fragmentation threshold
and the route from residue-evaporation to multifragmentation
They are observed both in peripheral and central collisions
LOPEZ/RIVET, WCI3 Town Meeting, College Station (Texas) February,