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Classical Drop Phase Diagram and Cluster Distributions

A. Chernomoretz, P. Balenzuela, C.O. DorsoDepartamento de Fisica, Universidad de Buenos Aires, Buenos Aires, Argentina.

(February 8, 2008)

The phase diagram, (T, ρ), of a finite, constrained, andclassical system is built from the analysis of cluster distribu-tions in phase and configurational space. The obtained phasediagram can be split in three regions. One, low density limit,in which first order phase transition features can be observed.Another one, corresponding to the high density regime, inwhich fragments in phase space display critical behavior of3D-Ising universality class type. And an intermediate den-sity region, in which power-laws are displayed but can not beassociated to the abovementioned universality class.

PACS number(s): 25.70 -z, 25.70.Mn, 25.70.Pq,02.70.Ns

I. INTRODUCTION

The multifragmentation phenomenon that takes placein nuclei with excitation energies above 2 MeV/nucleonhas been one of the central issues of the nuclear commu-nity during the past two decades. Many experimentaland theoretical efforts have been concentrated towardsthe understanding of the mechanisms involved in thatprocess. In particular, one feature that triggered the in-terest on the field was the important detected productionof intermediate mass fragments (IMF’s). Highly excitednuclei break up in many IMF’s in Fermi energy rangereactions. This feature, along with the fact that earlycalculations of caloric curves showed an approximatelyconstant behavior, was interpreted by many groups as asignature of a phase transition taking place in finite nu-clear systems [1–4]. In recent contributions the behaviorof the microcanonical heat capacity was used to experi-mentally characterize the transition as a first order one[6]. Within this picture, the latent heat can be associatedwith the transformation between a Fermi liquid and a gasphase, composed by light particles and free nucleons.

Despite that the familiar liquid-gas transition frame-work seems to be appropriate to deal with the nuclearcase, there are some peculiarities that are worth to beconsidered. One key point usually disregarded is that, byits own nature, multifragmentation in nuclear reactionsshould be a priori analyzed as a nonequilibrium process(see [7,8]). Even if an equilibrium scenario is adopted, amajor point is that we are dealing with a phase transitionoccurring in a finite system. Therefore, several thermo-dynamical features, e. g. the entropy extensivity, cannotbe taken for granted and the agreement between differ-ent statistical ensembles predictions cannot be invokedanymore [10]. A working hypothesis, usually made bystatistical models in order to apply a tractable global

equilibrium picture, is that a freeze-out volume can bedefined inside of which the existence of an equilibratedensemble of clusters can be assumed. In these models,the behavior of thermodynamics quantities is closely re-lated to the way the system is partitionated into clustersgiving rise to internal surfaces (isolated drops).

The aim of the present paper is to analyze the relation-ship between cluster distributions inside a fixed volumeand the respective thermodynamic description. In orderto do that a confined system that interacts via a two-body Lennard-Jones potential is studied. A moleculardynamics, MD, approach is used. The analogy betweenthe nuclear force and the van der Waals interaction sup-ports the use of this simplified classical model to obtainqualitatively meaningful results.

Given the MD microcanonical description, a connec-tion with thermodynamical quantities can be establishedvia well known principles from mechanics. For instance,the pressure and temperature can be estimated from thegeneralized equipartition theorem [11], whereas the spe-cific heat can be related to kinetic energy fluctuations[12]. On the other hand, a complementary analysis ofthe microscopic correlations at a “nucleon” level of de-scription can also be considered. To that end, differentfragment-recognition algorithms can be used in order tounveil different particle correlation properties.

Previous studies [13] have already dealt with this’cluster structure-thermodynamic description’ mapping.They made used of a cluster definition a la Coniglio-Klein, and were mainly focused on the system behaviorin the supercritical phase (ρ > ρc, T > Tc).

In this contribution we consider two alternative frag-ment definitions that, differently from Coniglio-Kleinclusters (that in the present paper will be called MSTEclusters, as will be explained in Sec. III), are built inwell defined and physically meaningful spaces. The firstone, associated with the so called minimum spanningtree fragment recognition method, is based on configu-rational information (MST clusters) [14]. The secondone uses complete phase space information in order todefine a fragment set according to the most bound den-sity fluctuation in phase space [16] (ECRA clusters). Us-ing this fragment characterization as a fundamental pieceof information, consistent phase diagrams are built fromscratch for the analyzed finite systems. In addition, therelationship between critical signatures in the fragmentdistributions and the corresponding system localizationin the phase diagram is established.

This paper is organized as follows. In Section II we willdescribe the model used in our simulations. Section III is

1

devoted to a characterization of the used cluster defini-tions. A description of expected signals associated withphase transitions in finite systems is given in Section IV.In Section V the study of phase diagram is presented.Finally, in Section VI, conclusions are drawn.

II. THE MODEL

The system under study is composed by excited dropsmade up of particles interacting via a 6-12 Lennard Jonespotential, which reads:

V (r) =

4ǫ

[

(

σr

)12

−(

σr

)6

−(

σrc

)12

+(

σrc

)6]

r ≤ rc

0r¿rc

(1)

We took the cut-off radius as rc = 3σ. Energiesand distances are measured in units of the potentialwell (ǫ) and the distance at which the potential changessign (σ), respectively while the unit of time used is:

t0 =√

σ2m/48ǫ.In order to constrain the dynamics we used a spherical

confining ‘wall’. The considered external potential be-haves like Vwall ∼ (r − rwall)

−12 with a cut off distancercut = 1σ , where it smoothly became zero along withits first derivative. The set of classical equations of mo-tion were integrated using the well known velocity Verletalgorithm [17], taking tint = 0.002t0 as the integrationtime step. Once the transient behavior was over a micro-canonical sampling of particle configurations every 5t0 upto a final time of 140000t0 was performed.

III. FRAGMENT DEFINITIONS

The simplest and more intuitive cluster definition isbased on correlations in configuration space: a particle ibelongs to a cluster C if there is another particle j thatbelongs to C and |ri − rj| ≤ rcl, where rcl is a parametercalled the clusterization radius.

We set rcl = rc = 3σ. The algorithm that recognizesthese clusters is known as “Minimum Spanning Tree”(MST). In this method only correlations in q-space areused, neglecting completely the effect of momentum. Aswas shown in Ref. [14], MST clusters give incorrect infor-mation about the meaningful fragment structure of thesystem for dense configurations. However, an interestingpoint to be notice is that it can still provide useful in-formation about the limit imposed by the constrainingfinite volume to the formation of well defined fragmentsin configurational space. As rcl = rcut, no inter-clusterinteraction exists, so cluster surfaces can be univocallydefined.

An extension of the MST is the “Minimum SpanningTree in Energy” space (MSTE) algorithm. In this case, a

given set of particles i, j, ..., k, belongs to the same clusterCi if:

∀ i ǫ Ci , ∃ j ǫ Ci / eij ≤ 0 (2)

where eij = V (rij)+ (pi −pj)2/4µ, and µ is the reduced

mass of the pair {i, j}. This cluster definition resem-bles the clusterization prescription adopted by Coniglioand Klein [18] and was the one used by Campi et al.

in Ref. [13]. The MSTE algorithm searches for config-urational correlations between particles considering therelative momenta of particle pairs, and typically recog-nizes fragments earlier than MST in highly excited un-constrained systems [14,15].

Finally, a more robust cluster definition is based onthe system “most bound density fluctuation”(MBDF) inphase space [16]. The MBDF is composed by the set ofclusters {Ci} for which the sum, E{Ci}, of the fragmentinternal energies attains its minimum value:

E{Ci} =∑

i

ECi

int

with ECi

int =∑

j∈Ci

Kcmj +

∑

j,k∈Cij≤k

Vj,k (3)

Kcmj is the kinetic energy of particle j measured in

the center of mass frame of the cluster which containsparticle j, and Vij stands for the inter-particle potential.

The algorithm that finds the MBDF is known asthe “Early Cluster Recognition Algorithm” (ECRA). Itsearches for simultaneously well correlated structures inboth, q, and p space, via the minimization of the poten-tial and the kinetic terms respectively.

The ECRA algorithm has been used extensively inmany studies of free expanding fragmenting systems[7,19] and has helped to discover that excited drops breakvery early in the evolution. In addition, in a recent contri-bution [8] it was shown that ECRA clusters are also suit-able to describe the fragmentation transition that takesplace in volume constrained systems.

IV. CHARACTERIZING THE TRANSITION

As mentioned in the introduction, several observablescan be studied in order to analyze phase transitions phe-nomena occurring in finite systems. They usually involvethe behavior of caloric curves [2,3,20–23], specific heats[6,24], kinetic energy fluctuations [4,5], fragment massdistributions [25,26], and critical exponents [26].

A. Thermodynamical Description

One of the most relevant quantities in the study offragmenting systems, either constrained or free to ex-pand is the caloric curve (CC). The CC is defined as the

2

functional relationship between the system energy andits temperature, given by:

T (E) =2

3(N − 1)< K >E (4)

being N the number of particles, and < K >E the meankinetic energy averaged over a fixed total energy MD sim-ulation [9].

In Figure 1(a) the CC’s are shown for the followingdensities: ρh = 0.55σ−3, ρc = 0.35σ−3, ρm = 0.07σ−3,and ρl = 0.01σ−3.

-4 -3 -2 -1 0 1 2 30

1

2

tem

pera

ture

(ε 0)

-4

-3

-2

-1

0

V (

ε 0)

-4 -3 -2 -1 0 1 2 3

energy (ε0)

0

0.05

0.1

σ V,K

(ε 0)

(a)

(b)

(c)

FIG. 1. The caloric curve is shown in panel (a). In pan-els (b) and (c) the potential energy (V ) and its root meansquared deviation as a function of the system total energy aredisplayed in panels (b) and (c) respectively. The followingdensity values: ρ = 0.01, 0.07, 0.35, and 0.55σ−3, displayedwith full triangles, diamonds, squares, and circles respectively,were considered.

Different behaviors can be recognized. For the morediluted case, ρl (full triangles), the corresponding CCdisplays a loop which ends in a linearly increasing tem-perature line, which we refer as vapor branch. Takinginto account that, within the microcanonical ensemble,the specific heat is defined as:

1

C=

∂T

∂E= −T 2

∂2S

∂E2(5)

it is clear that for this case negative values of C willbe found in the range 0.1ǫ0 < E < 0.6ǫ0. A negativevalue of the derivative of the temperature as a functionof the energy reflects an ’anomalous’ behavior of the sys-tem entropy for the respective energy range. A convex in-

truder in S(E), prohibited in the thermodynamical limit,arises as a consequence of the finiteness of the system andthe corresponding lack of extensivity of thermodynami-cal quantities like S. This signal is expected in first orderphase transitions, and is associated to a negative branchof the heat capacity between two poles [10]. Taking intoaccount the relationship between kinetic energy fluctua-tions and the system specific heat [5,29]:

1

N< (δK)2 >E=

3

2β2(1 −

3

2C) (6)

it can be seen that negative values of the specific heatappear whenever (δK)2 get larger than the canonical ex-pectation: 3N

2β2 . This behavior has already been verified

in Ref. [8].As the density is increased the loop is washed away and

it is replaced by an inflection point (ρm, full diamonds inFig.1(a)). This corresponds to the merging of the specificheat negative poles into a single local maximum. Finally,for higher densities, ρc and ρh (filled diamonds, and cir-cles), no major changes in the CC’s second derivative isobserved.

In Fig. 1(b) the mean potential energy per particle,V , as a function of the total system energy is displayed.For the lowest densities two different regimes can be rec-ognized. First, a steep increasing behavior for energiesE < 0.5ǫ0, that can be related to an increase of themean interparticle distance and the appearance of inter-nal surfaces, as the number of atractive bonds decreases.Second, a saturating behavior for larger energy values,E > 0.5ǫ0. For more dense systems, no such feature canbe observed. A smooth behavior is displayed by V .

Another interesting feature can be noticed looking atthe root mean square deviation of partial energies (ki-netic or potential), σV,K , shown in Fig. 1(c). For thelowest densities (filled triangles and diamonds) an abruptdecrease is observed at E ∼ 0.5ǫ0, whereas for the high-est considered density, ρh = 0.8σ−3(filled circles), notrace of such behavior can be reported. For future refer-ence, it is worth noting that the transition in the mono-tonic character of the curve takes place at a density valueρ ∼ ρc = 0.35ǫ0(filled squares).

To attain a better understanding of the behavior ofthe magnitudes displayed in Figs. 1(b), and (c), we havefollowed two strategies: i) to study particle-particle cor-relations, ii) to analyze fragment mass distributions ac-cording to the already presented fragment definitions. Inwhat follows we present results obtained within the firstapproach, and postpone to the next subsection the anal-ysis of fragment properties.

In Fig.2 the distribution of interparticle distances, rij ,is shown. Two density values already considered in Fig1,

3

namely ρh and ρl, are shown in panels (a) and (b) re-spectively.

For each density, three total energy values, lowerthan(solid line), equal to (dotted line), and larger than(dashed line) E = 0.5ǫ0, were considered (see caption).In both panels, a vertical line was included indicating theinteraction cut-off radius.

0 10 20 300

0.02

0.04

0.06

frac

tion

pair

s

0 1 2 3 4 5 6 7

r (σ)

0

0.1

0.2

frac

tion

pair

s

(a)

(b)

FIG. 2. Normalized interparticle distance distribution cal-culated for ρl and ρh systems are shown in panels (a) and(b) respectively. The following three total energy values,E = 0.0, 0.5, 1.0ǫ0, were considered in panel (a), whereasE = −0.50, 0.5, 1.0ǫ0 for panel (b). They are shown with,solid, dotted, and dashed lines respectively. Note the differ-ent scales used for rij in both panels.

In panel (a) (very diluted case) a well defined structureof interacting particle pairs can be seen at low energies(continuous line). The displayed peaks signal the pres-ence of a rather large self-sustained drop, with a first,second and even third nearest neighbors structure. Asenergy increases, this structure fades away, and an im-portant reduction of the number of interacting particlescan be noticed. This occurs as a consequence of the spa-cious volume available , that is big enough to accommo-date small non-overlapping particle aggregates. In thistransition, well defined q-space clusters appear, surfacesare produced and V tends towards a residual value (seebellow).

A completely different behavior is observed for thedenser instance, ρh, shown in Fig. 2(b). In this case, nochanges in the statistical distribution of rij is observedas the energy is increased. (Note that the bell shapeddistribution associated with the ‘trivial’ noninteractingpair counting is superimposed over the peak structure.However, the presence of a first, second, and even a thirdnearest neighbors can still be traced). The container im-poses a severe volume restriction on the system, and even

for high total energy values, each particle is confined inthe attractive concavity of partners potential, betweenthe repulsive core of nearest neighbors.

This last observation can complete a general picture,within which the behavior of the mean potential energy,V , for the high density cases can be easily understood. Asmore energy is added to the system no structural tran-sition is allowed by the constraining volume. Particlescan not escape from neighbors most attractive potentialrange and a smooth increased in V , related to the aver-age time spent in the most negative potential areas, canbe seen (Fig 1(b)).

The decreasing of σV , observed for low densities (filledtriangles, and diamonds in Fig. 1(c)), can be associatedto the loss of ‘attractive bonds’ between particle pairsthat takes place when a non interacting light clusterregime dominates. As an increasing number of particlesstop interacting with one another, the system dynamicsgets ’less chaotic’ (see [28] for a dynamical characteriza-tion of this system), and a decrease of σV is induced.

On the other hand, for high densities, the region withthe strongest non-lineal nature becomes the most rele-vant part of the interaction potential. As a consequenceof that, an increase of the fluctuations in potential en-ergy between successive configurations as total energy isadded can be expected (see the behavior displayed by σV

in Fig 1(c)).It is worth noting that the presented picture can also be

used to interpret recent results regarding the behavior ofthe maximum Lyapunov exponent, MLE, in constrainedsystems (see [28]). Moreover, a striking similitude be-tween the behavior of σV (E, ρ) and MLE(E, ρ) can benoticed, comparing Fig.1(c), and Fig.5 of Ref. [28].

B. Fragments Inside the Volume

In section III, three fragment recognition algorithmswere described. Each one of them make use of differentcorrelation information in order to link particles into clus-ters, and then give different physical information aboutthe system under study.

4

0.001

0.01

0.1

1

10

0.001

0.01

0.1

1

10

Yie

ld

1 10 100

mass

0.001

0.01

0.1

1

(a)

(b)

(c)

FIG. 3. MST cluster mass distributions calculated for den-sities: ρ = 0.01, 0.03, and 0.10σ−3 are displayed in panels (a),(b), and (c) respectively. Three total energy values were con-sidered for each density, Etot = 0.0, 0.5, and 1.0ǫ0.They aredisplayed as full, dotted and dashed lines respectively

In Fig. 3 the results of the MST algorithm analysisare shown. MST spectra were calculated for differentenergies (E = 0.0, 0.5, and 1.0ǫ0, displayed as full, dottedand dashed lines respectively) for three system densities:ρ = 0.01, 0.03, and 0.10σ−3. They are shown in panels(a), (b), and (c), of Fig.3 respectively.

At low densities, panel 3(a), the system evolves froma heavy cluster dominated behavior at low energies, to-wards a light cluster dominated behavior, at high ener-gies. This reflects the fact that the constraining volumeis large enough to allow the system to fragment in welldefined drops as energy is increased. It is interesting tonotice that results displayed in Fig. 3(a) correspond tofull triangle symbols in Fig.1(a), i.e. the one for which theCC displays a loop. It was argued that big fluctuationsin kinetic energy should be expected. This is indeed thecase and it is related to the fact that well defined surfacesappear in the system (see [8]).

On the other hand, as density increases, the mass dis-tributions converge to a u-shaped one. Almost no spa-tially well separated structures (i.e. MST clusters) canbe identified, aside from the trivial huge cluster that com-

prises almost all the system particles.In Fig. 4, the effect of taking into account momen-

tum correlation in the definition of clusters is displayed.There, a cluster analysis for a system of N = 147 par-ticles with ρ = 0.2σ−3 using ECRA, MSTE, and MSTprescriptions is presented. The mass distribution func-tion, and the fragment internal energy as a function of thecluster mass, Eint(s), are shown in the first and secondcolumns, respectively. Three total energy values wereconsidered: Etot = −0.6 (panels (a)-(b)), 0.0 (panels(c)-(d)), and 0.5ǫ0 (panels (e)-(f)). Empty triangles, solidsquares, and empty circles correspond to MST, MSTE,and ECRA results respectively.

0 50 100 150 0 50 100 150

-1

-0.5

0

1 10 100

Yie

ld (

arb.

uni

ts)

0 50 100 150-1.5

-1

-0.5

0

1 10 100mass

0 50 100 150mass

-1.5

-1

-0.5

0

0.5

(a) (b)

(c) (d)

(e) (f)

Eint (ε

0 )

FIG. 4. Fragment mass distribution, and cluster internalenergy values as a function of the cluster mass, are shown inthe first and second columns respectively. Three system totalenergies are considered: Etot = −0.6ǫ0, panels (a-b), 0.0ǫ0,panels (c-d), and 0.5ǫ0, panels (e-f). Empty circles, filledsquares, and empty triangles, correspond to ECRA, MSTE,and MST data, respectively.

It can be seen that, for this density, the MST pre-scription finds just a big drop. On the other hand, both,the MSTE and ECRA algorithms, find non-trivial clusterdistributions that show the expected transition from a u-shaped, towards and exponential behavior, as the systemenergy is increased. As a general rule, ECRA algorithmproduces more bound clusters than the MSTE one. Thisis consistent with the claim that the MBDF are found bythe ECRA algorithm. Due to this feature, from now on,

5

just the properties of the system according to its ECRAcluster structure will be considered.

C. Transition Signals

As seen in Fig.4, in which a system at a ρ = 0.2σ−3

density was considered, the ECRA cluster distributionsundergo a transition from a U-shaped spectrum towardsan exponentially decaying one. Moreover, the one corre-sponding to the intermediate energy value 4(b), displaysa power law like behavior. In fact, for any other densitystudied, the same behavior can be reported, and a totalenergy value can be determined, for which a power-lawlike mass distribution can be found.

This result is interesting, because it implies that somekind of transition between two regimes, one with, andother without large ECRA drops, i.e. with or without thepresence of liquid-like structures, is taking place in highdensity systems. Moreover, this transition can not bedetected simply using spatial correlations information. Inthis way, considering the appropriate cluster prescription,power-law mass distributions can be used to trace thetransition line in a (T, ρ) diagram.

Within the Fisher Droplet Model [34], when dealingwith infinite systems, one expects to find such a scale-free distribution at a critical point, where a continuoustransition takes place, and scaling hypothesis can be ap-plied. However, it has been reported [23,30,31] that sev-eral critical signatures also appear when first order phasetransition are analyzed in small systems. In particular, asstated in Ref [31], in small systems the largest cluster getsa size comparable to the vapor fraction before dissapear-ing when the system crosses the coexistence line. There-fore there is an energy for which a pseudo-invariance ofscale and a resemblance with critical behavior can beexpected for small systems undergoing phase transitionsthat, in the thermodynamical limit, would be univocallyclassified as first order ones.

In order to search for a power-law behavior in fragmentdistributions the following single parameter function wasemployed to fit fragment mass spectra. The contributionof the largest fragment was disregarded [26]:

n(A) = q0A−τ ,with q0(τ) =

1∑

A A1−τ(7)

n(A) is the number of fragments with mass number A,and q0 is a normalization constant.

The quality of the fitting procedure was quantified us-ing the standard χ2 coefficient (see [32] for details), andan energy value, E∗, was associated to the best fittedspectra. In Fig.5(a) a typical χ2 calculation is shown forECRA clusters in a ρ = 0.10σ−3 system. From the figure,a value of E∗ = 0.10 ± 0.05ǫ0, can be reported.

0.1

1

10

χ2

1

1.5

2

2.5

M2

-0.6 -0.3 0 0.3 0.6 0.9

energy (ε0)

0

0.01

0.02

0.03

0.04

NV

M

(a)

(b)

(c)

FIG. 5. Transition signatures calculated for a ρ = 0.10σ−3

system. The χ2 fitting coefficient, the second moment of themass distribution, M2, and the normalized mean variance ofthe largest fragment, NVM, are displayed in panels (a), (b),and (c) respectively.

Another useful observable to search for scale-free dis-tribution functions is the second moment of the clusterdistribution M2 =

∑′A A2n(A). As in the determination

of τ , the largest cluster is excluded from the primed sum.M2 is proportional to the compressibility κT , that, in thethermodynamical limit, presents a power-law singularityat a second order transition. For finite system the di-vergence is replaced by a maximum. In panel 5(b) it isshown the estimation of M2(E), for a ρ = 0.10σ−3 sys-tem. It can be noticed that the maximum is located atthe energy value for which the spectra is best fitted by apower-law like dependence.

This kind of agreement is also achieved when the nor-malized mean variance, NVM, of the largest fragmentmass, Amax, is analyzed. NVM is defined as:

NV M =< A2

max > − < Amax >2

< Amax >(8)

and it proved to be a robust tool for the characteriza-tion of transition phenomena in which an enhancement offluctuations occurs [33]. As can be seen from panel 5(c),this signal, although slightly shifted within the workingresolution, also peaks in the same region of the previousones.

The signal agreement reported for the ρ = 0.10σ−3

case in Fig.5, is also achieved for every density analyzedin this paper. This means that, in the whole density

6

range, power-law mass distributions for phase space de-fined ECRA clusters are found whenever large fluctua-tions take place in the system.

In order to properly characterize the state of the sys-tem which displays power-law mass distributions, the val-ues attained by the τ exponents must be analyzed, as afunction of the density. This dependence is shown inFig.6(a). For a true critical phenomenon, taking placein three dimensional systems, 2 ≤ τ ≤ 3 is expected[34]. This condition is not satisfied for systems withρ < 0.05σ−3. For this highly diluted systems, the ob-served free-scale distribution is not expected to survivethe thermodynamic limit. It is not related to any con-tinuous transition, but arises as a finite-size effect. It isworth noting that ρ ∼ 0.05σ−3 is the maximum densityfor which the respective caloric curve displays a loop, andthen, negative cv (see Fig.1).

0 0.2 0.4 0.6 0.8 1

ρ (σ-3)

0

0.5

1

1.5

2

γ

(b)

2

2.5

3

3.5

τ

(a)

FIG. 6. Estimated values from ECRA cluster distributionsfor the critical exponents τ and γ are shown in panels (a),and (b) respectively. The expected values for the 3D-Isinguniversality class are displayed by a dashed line.

On the other hand, for ρ > 0.05, the calculated τ ex-ponents show a rather good agreement with the τ = 2.21value (dashed line), expected for liquid-gas transitions.

In a second order phase transition, the behavior of M2

near a critical point can be described in terms of thecritical exponent γ [26]:

M2(ǫ) ∝ |ǫ|−γ . (9)

where, ǫ = (Ec−E)/Ec, measures the distance to thecritical point. This relation is valid for an infinite systemin the limit ǫ → 0. As already mentioned, in a finite sys-tem M2 displays a maximum instead of the divergence

predicted in equation (9). Having this in mind, a cal-culation procedure introduced in Ref. [27] (γ-matching),that takes care of finite size effects, was used to calculatethe γ exponent value for our system (see also Ref. [32]for details).

The results of the γ exponent estimation is shown inFig.6(b). The obtained values tends toward γliq−gas =1.23 (dashed line) value, that is expected for a liquid-gas transition. It is worth noting that this convergenceis achieved for densities ρ ≥ 0.35ǫ0 (this value equalsρc in Fig.1). For this density, a change in the behaviorof the potential energy fluctuations as a function of thetotal energy was reported in Fig.1(c). This can be as-sociated (see Sec.IVA) to the onset of the invariance ofthe statistical interparticle distance distribution (Fig.2),that takes place as a consequence of the imposed volumerestriction, and precludes the occurrence of a continuoustransition.

V. PHASE DIAGRAM

In the previous section several signatures of a change inthe properties of ECRA fragment mass distributions wereanalyzed. At any given density, a system energy, E∗(ρ),can be determined for which large fluctuations appear inthe system, and a scale-free like mass distribution can befound.

0 0.2 0.4 0.6 0.8

ρ (σ-3)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

E* (

ε 0)

FIG. 7. Density dependence of the system energy, E∗, atwhich transition signals are detected. The dashed line is in-cluded to guide the eye.

In Fig.7, the dependence of E∗ with the system den-sity can be seen. It can be noticed that for ρ ≥ 0.35σ−3,a rather constant value is attained for the transition en-ergy. A similar result was reported in Ref. [13], using thealready presented MSTE cluster definition.

Some insight about the properties of the system alongthe line depicted in Fig.7, can be gained from the analy-sis of the interplay between the mean internal potentialenergy per particle stored in ECRA clusters, Vint, andthe mean inter-cluster interaction energy, Vic.

7

Vint =∑

i<j

i,j∈Ck

Vi,j (10)

Vic =∑

i∈Ckj∈Clk 6=l

Vi,j (11)

Vint stands for the mean potential felt by a particle ina cluster due to its interaction with the other membersof the same cluster. On the other hand, Vic representsthe mean interaction potential felt by a particle due toits interactions with particles belonging to the other clus-ters. This magnitudes are displayed in Fig.8 for densitiesρ = 0.01, 0.20, 0.50, and 0.80σ−3 in panels (a) to (d) re-spectively.

-1

-0.5

V (

ε 0) Vint

VinterClus

-3

-2

-1

V (

ε 0)

-3

-2

-1

V (

ε 0)

-1 -0.5 0 0.5 1

energy (ε0)

-3

-2

-1

V (

ε 0)

(a)

(b)

(c)

(d)

FIG. 8. Vint (empty circles) and Vic (empty squares) asa function of the system total energy are shown. Densities:ρ = 0.01, 0.20, 0.50, and 0.80σ−3 are displayed in panels (a),(b), (c), and (d), respectively

The general trend displayed by Vint and Vic is easyto understand. At any given density, for low energies, abig ECRA cluster can be found having a large bindingenergy. In addition, as no many other clusters beside thebiggest one exist, Vic ∼ 0. At high energies, the ECRA

partitions turn out to present a rather high multiplicityof light clusters. Consequently Vint → 0 in this limit andVic absolute value increases.

After a close inspection of Fig.8, one can realize that,for any given density, the energy at which Vint = Vic, hap-pens to be E∗(ρ), i.e. the energy at which power-law likemass distributions appear. This means that scale-freeECRA mass distributions have the following property:a balance is established between the mean potential en-ergy a particle, that belongs to a given cluster, feels dueto its interaction with other members of the same clus-ter and the one associated with its interaction with therest of the particles in the system. This potential energybalance, that does not allow to distinguish contributionsfrom inside or outside of a cluster, is reminiscent of thevanishment of the chemical potential and surface tensionterms that takes place at the critical point within theFisher droplet model framework [34].

Gathering all the information obtained so far, thephase diagram for our 147-particle Lennard-Jones dropcan be built. In Fig.9 the resulting phase diagram ispresented. The empty circles are (T∗, ρ) points obtainedfrom the aforementioned cluster analysis. The full line isan estimation of the coexistence line, as obtained fromthe analysis of the system specific heat, cv (see Eq.6). Inthis case, for densities for which the respective CC dis-plays a loop, T (E∗) has been taken as the average tem-perature between the two values corresponding to thelocation of the cv singularities. For denser cases, thelocation of the maximum of the cv has been taken. Noreliable estimation of the energy that maximizes cv couldbe achieved for ρ > 0.35σ−3.

0 0.2 0.4 0.6 0.8

ρ (σ-3)

0

0.5

1

1.5

2

2.5

3

tem

pera

ture

(ε 0)

A B C

FIG. 9. Phase diagram for a 147 particle Lennard Jonesdrop. Cluster-based and cv-based determination of the coex-istence line is marked with empty circles, and full line respec-tively. The dashed line is included to guide the eye.

From the figure, it can be seen that the symmetry be-tween low and high densities, characteristic for the infi-

8

nite size limit, is lost in the finite size system. An almostlinear raising branch, for ρ > ρc, can be seen instead. Forevery density value, below the depicted coexistence line,a liquid-like ECRA structure can be identified, whereasa vapor behavior can be observed above it.

It is interesting to notice that three density regionscan be identified. The region labelled A (ρ ≤ 0.05σ−3)presents the signals expected for a first order phase tran-sition occurring in a finite system: the correspondingcaloric curve shows a loop that can be associated to anegative specific heat, and a structural transition can berecognized (Fig.2(a)). These region is the only one inwhich the available volume is large enough for the sys-tem to fragment into a set of non-overlapping drops.

On the other hand, in region C (ρ > 0.35σ−3), wherethe system is rather compressed, and MST algorithm rec-ognizes just one big fragment, a second order transitionseems to occur in phase space, whenever the transitionline is crossed. No anomaly in the respective caloric curveis observed, no qualitative changes in the configurationstatistical properties is reported (Fig.2(b)), and the cal-culated critical exponents, τ , and γ, are in good agree-ment with 3D-Ising (liquid-gas) universality class.

In between those two regions, region B (0.05σ−3 < ρ <0.35σ−3) can be identified. The finite size of the systemplays a major role in this density range. Even a sensi-ble τ value suggests that physically meaningfull scalingproperties are present in the system, the correspondingγ exponent values are too low to classify the transitionas a continuous one.

VI. CONCLUSIONS

In this paper we have undertaken the analysis of ther-modynamical properties of a finite classical system con-fined in a volume. Apart from the basic interest on sucha problem, it might be relevant on the frame of the de-scription of fragmenting system according to statisticalmodels.

We have been able to find, that from a coordinate spacestand point the equation of state of such a system isquite simple. There is a maximum value of the density(ρ ∼ 0.05σ−3) up to which the system can undergo afirst order phase transition. In this region, the associatedCC displays a loop and the thermal response functionattains negative values. This feature allowed us to definea transition curve. For higher densities, there is simply noroom enough to allow the system to develop well definedinternal surfaces, i.e. only one big drop can be detected.

When one turns into a description of fragments inwhich correlations in p-space are taken into account, amuch reacher structure appears. Now, even for densitiesbigger than ρ ∼ 0.05σ−3 transitions from u-shaped massspectra to exponentially decaying ones are signaled bythe appearance of scale-free mass distribution of ECRAclusters. However, from the analysis of critical expo-

nents, τ and γ, related to the displayed power-law distri-bution, a further classification in density ranges can beobtained.

In the abovedefined region A, the obtained τ values aretoo big to be related to a true critical behavior. For den-sities 0.05σ−3 < ρ < 0.35σ−3, eventhough the values ofτ exponents are in good agreement with the correspond-ing 3D-Ising universality class, the value attained by γcritical exponents came out to be too low. Finally, fordensities above ρ ∼ 0.35σ−3, both, τ and γ, are quiteclose to the accepted values for the 3D-Ising universalityclass.

This is of particular interest for the statistical modelapproach because by definition, at freeze out, an ensem-ble of well defined fragments is to be constructed. Suchassumption implicitly locates the system under study inregion A, where first order phase transition is to be ex-pected. It is then clear that in the frame of such ap-proaches regions B and C are excluded from the analysis.

A natural continuation of this work is to analyze the re-lation between fragmentation inside the container volumeand the corresponding asymptotic mass spectra whenwalls are removed. This work is currently under progress.

Finally, we would like to mention that the behaviorof the equation of state for small systems has been un-dertaken in Ref. [13]. The results obtained in that workdiffers from ours (critical exponents) but they were basedon properties of MSTE fragments, which we have disre-garded favoring the ECRA approach because of the rea-sons stated right after Fig.4.

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