entropy, correlations, and ordering in two dimensions

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 7 15 AUGUST 2000

Entropy, correlations, and ordering in two dimensionsF. SaijaIstituto Nazionale per la Fisica della Materia (INFM), Unita` di Ricerca di Messina, Italy

S. PrestipinoIstituto Nazionale per la Fisica della Materia (INFM), Unita` di Ricerca di Messina, Italy and InternationalSchool for Advanced Studies (SISSA-ISAS), I-34013 Trieste, Italy

P. V. Giaquintaa)

Istituto Nazionale per la Fisica della Materia (INFM), Unita` di Ricerca di Messina, Italy andDipartimento di Fisica, Universita` degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy

~Received 16 March 2000; accepted 17 May 2000!

The ordering of simple fluids in two dimensions was investigated using the residual multiparticleentropy~RMPE! as a measure of the relevance of correlations involving more than two particles inthe configurational entropy of the system. To this end, we performed Monte Carlo simulations oftwo prototype systems, i.e., Lennard-Jones particles and hard discs. Consistent with previousstudies, we found that, on approaching the freezing transition, the RMPE of the fluid undergoes achange from negative to positive values. However, in two dimensions the vanishing of the RMPEappears to be more directly related to the formation of six-fold orientationally ordered patches, aprocess which foreshadows the freezing transition. The specificity of the structural conditionattained by the fluid in a state corresponding to a vanishing RMPE was further corroborated by ananalysis of the shape of the radial distribution function~RDF!: in fact, it turns out that the spatialprofiles of the RDF of the Lennard-Jones fluid along a zero-RMPE locus can be superimposed atmedium and large distances notwithstanding the difference of density and/or temperature of thecorresponding thermodynamic states. The same long-range profile of the RDF is shared also by harddiscs in the cited condition. Such a ‘‘scaling’’ property also holds in three dimensions where itprovides a suggestive nexus between the ordering criterion based on the vanishing of the RMPE andthe Hansen–Verlet freezing rule. ©2000 American Institute of Physics.@S0021-9606~00!51331-2#

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I. INTRODUCTION

The entropy of an even partially disordered materialfolds information on the state of the system which lies byond the strictly thermodynamic level. In principle, this iformation can be extracted by resorting, in a classistatistical–mechanical framework, to a ‘‘representation’’the entropy in terms of the complete hierarchy ofn-pointdistribution functions.1 In general, this is rather obviouslyprohibitive task. A few years ago, Giaquinta and Giuntacussed instead on a cumulative ‘‘measure’’ of the statistweight associated with correlations involving at least thparticles, the so-called residual multiparticle entro~RMPE!.2 This quantity is obtained by subtraction of th‘‘pair entropy,’’ i.e., the integrated contribution associatwith the pair distribution function only, from the excess etropy of the fluid. Indeed, the pair entropy, while being quatitatively preponderant with respect to other terms, isnegative–definite quantity~whatever the system or the themodynamic state! which does not convey any releva‘‘qualitative’’ information on the equilibrium structure of thsystem. On the contrary, the RMPE does not have a defisign. The crossover undergone by the RMPE from negato positive values was originally put in relation with th

a!Author to whom correspondence should be addressed; [email protected]

2800021-9606/2000/113(7)/2806/8/$17.00

Downloaded 30 Oct 2008 to 147.122.3.100. Redistribution subject to AIP

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freezing transition of hard spheres.2 Since then, a similaranalysis has been extended to a variety of mopotentials,3–8 including mixtures,9–12 systems composed ononspherical particles,13,14 and lattice gases.15,16 In all casesexamined so far, the vanishing of the RMPE proves to beintrinsic signature of the incipient ordering of the fluid,condition which faithfully heralds the occurrence, at neardensities and/or temperatures, of a phase transition inmore structured phase. In this respect, the zero-RMPE cdition can be used as a one-phase ordering criterion whhas been successfully tested against such diverse thermnamic phenomena as freezing, fluid–fluid phase separanematic ordering, and the Kosterlitz and Thouless metinsulator transition in a two-dimensional~2D! Coulomb lat-tice gas.

Many semiempirical rules have been proposed withaim of locating, say, the transition from the liquid to the sophase, without resorting to the knowledge of the free eneof the two phases. Perhaps, the very first of such proposathe melting criterion proposed by Lindemann,17 which statesthat in a solid the ratio of the root-mean-square displacemof a particle to the average nearest-neighbor distance is a0.15 at the melting point. On the other hand, the mostmous freezing criterion is that due to Hansen and Verwho observed that the height of the first peak of the structfactor,S(k), is about 2.85 along the freezing line. This ‘‘unil:

6 © 2000 American Institute of Physics

license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

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2807J. Chem. Phys., Vol. 113, No. 7, 15 August 2000 Ordering in two dimensions

versal’’ feature has been verified by both numerical simution and experiment for a wide class of three-dimensio~3D! systems.18 Just to give a few more examples, we recstill another empirical rule, proposed by Wendt and Abham, in terms of the ratio between the heights of the fiminimum and of the first maximum, respectively, of the rdial distribution function~RDF!.19 In many 3D liquids, thisratio is about 0.14 at freezing. On the dynamical side, Lo¨wennoted that in a liquid with a Langevin dynamics the ratiothe long- to the short-time diffusion coefficient is about 0at the freezing point.20 More recently, Malescio and coworkers introduced a criterion based on the numerical inbility of the iterative solution of the Ornstein–Zernikequation.21

As a matter of fact, most of the above criterions aspecific of a given transition. Furthermore, they may noteasily transferred, without modifications, to 2D systewhere the crystalline phase is unstable at all temperatureT,save forT50.22 As a result, the Lindemann rule is uselessits original formulation.23 All of the other criterions, whilesuitable in principle for 2D systems, are not valid in geneand must be modified. For instance, in 2D the height offirst peak of the structure factor at freezing is much higthan 2.85; moreover, its precise value is more sensitive toshape of the interaction potential than in 3D, suggestingthe Hansen–Verlet rule cannot be extended to arbitrarymensionality in a simple way.24 Lowen verified the validityof his dynamical criterion also for 2D soft-sphere fluids,still using a Langevin equation for the dynamics. Again, tcritical ratio of the two diffusion coefficients is close to 0~actually, a bit smaller than that!.25 However, the real prob-lem with this criterion is that it ceases to be valid whendifferent dynamics is considered, e.g., the Newtonian osince in this case only one single diffusion coefficient candefined.26

The 2D case is also particularly challenging in viewthe many different scenarios that have been proposed fofreezing transition,27 including a two-stage continuous transition from the fluid to the solid phase which would taplace through the formation of a hexatic, bonorientationally-ordered phase. Such a possibility was contured in the celebrated KTHNY theory.28

However, we do not aim here at investigating in detthe nature of the phase diagram of specific 2D models.know that this is a difficult task to handle numerically evin simple cases.29,30Instead, we intend to ascertain the natuand sensitivity of the indications given by the RMPE in rlation to latent tendencies to ordering, even of an unconvtional type as that exhibited in the hexatic phase. Actuathis phase may ultimately prove to be metastable. Howeeven in such a case, signatures of angular ordering shshow up in the dense fluid close to freezing and becomanifest in the RMPE as well. In order to elucidate thaspect, we have chosen, as established prototype moLennard-Jones~LJ! particles and hard discs~HD!.

This article is organized as follows: In Sec. II we intrduce and discuss the RMPE. Section III is devoted to ascription of the models and of the method. In Sec. IV,present the results of this study with attention given to


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characterization of angular order in terms of the orientatiocorrelation function~OCF!. A rationalization of the zero-RMPE condition as an indication of the incipient orderinga fluid is further discussed in Sec. V in terms of the scalbehavior of the RDF. Section VI is finally devoted to cocluding remarks.

II. THE RMPE: A BRIDGE BETWEEN ENTROPY ANDCORRELATIONS

The statistical entropy of an open~i.e., grand-canonical!system can be expanded as an infinite series:1

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wheres(ex) is the excess entropy per particle in units of tBoltzmann constant,kB , andsn is then-body entropy that isobtained from a suitable resummation of spatial correlatibetween up ton particles. An analogous expansion holdsthe canonical ensemble,15 with an identical expression fothe partial quantitiessn in terms of reduced distribution functions. In particular, the pair entropy per particle reads

s2521

2rE dr @g~r !ln g~r !2g~r !11# , ~2!

wherer andg(r ) are the number density and RDF, respetively. The RMPE is defined as

Ds5s(ex)2s2 . ~3!

At variance with the pair entropy,Ds exhibits a nonmono-tonic behavior as a function of eitherr or T. In particular, itis negative at low density or high temperature, and becompositive as a more ordered phase is approached. A posRMPE implies aslowing downin the reduction of the con-figurational space accessible to the system, caused by acreasing volume and/or temperature. This effect, that is scifically associated with the inversion of trend of the RMPis the outcome of theincreaseof the number of states that, ithe overall entropic balance, can be associated to correlatof order higher than two. This increase is obviously relatto the systematically negative background level set bypair entropy. Such a feature is the underlying fingerprint onew structural condition that is being built up by the syste‘‘forced’’ to exploit a different form of aggregation by compelling thermodynamic constraints. Not at all surprising, tRMPE indicates that such a structural, cooperative rearranment is ascribable to many-particle correlations.

The RMPE phenomenology that has been exploredfar regards mostly first-order phase transitions in 3D. Rcently, the connection between the vanishing of the RMand the ordering of the fluid has also been verified in 2even for long-ranged particle interactions.16 In fact, it hasbeen shown that the RMPE vanishes in a Coulomb latgas along the infinite-order Kosterlitz–Thouless transitline, which separates the insulating from the conducting flphase. This somewhat surprising circumstance proves thetreme sensitivity of the RMPE to all structural and thermdynamic changes which occur in a disordered system. Wsuch premises, it is natural to ask what is the behavior of


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2808 J. Chem. Phys., Vol. 113, No. 7, 15 August 2000 Saija, Prestipino, and Giaquinta

RMPE in a 2D continuous fluid system, where the reducdimensionality may actually decouple the onset of orientional order from the long-ranged positional one, thus caing unconventional cooperative phenomena to occur asinstance, the emergence of a hexatic phase.

III. MODELS AND METHOD

We performed Monte Carlo~MC! simulations, in thecanonical ensemble, of a system of 450 particles interacin 2D through a LJ potential:

uLJ~r !54eF S s

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2S s

r D 6G , ~4!

wherer is the interparticle distance. The potential was trucated atr 52.5s, and the usual correction was made to tainto account the effect of the neglected tail on the thermonamical quantities.18 The same technique was also usedsimulate a system of 450 hard discs. Particles were encloin a square box with periodic boundary conditions. The itial configuration was always that of a perfectly ordered hagonal crystal. The equilibration of the samples typicatook (23105) – (13106) MC cycles, depending on the density. A cycle consisted of one attempt to sequentially chathe position of all the particles. The maximum displacemof a particle was tuned, during the run, so as to keepacceptance ratio of the MC moves as close to 0.5 as possThe relevant thermodynamic averages were computed(43105) – (73105) MC cycles. The RDF histogram waconstructed with a spatial resolutionDr 5s/20, and updatedevery 100– 500 MC cycles. The RDF was evaluated upRmax5L/2, whereL5(N/r)1/2 is the simulation-box lengthAt a distance of orderRmax, the RDF was never found todiffer significantly from unity. The fulfillment of this condi-tion is important since, otherwise, a bad estimate of theentropy would be obtained, owed to the fact that, inhigh-density regime, several peaks of the RDF beyondfirst give a sizeable contribution to the integral in Eq.~2!.

In the case of the LJ system, we spanned a wide denrange at two supercritical temperatures,T* 50.7 and 1,whereT* 5kBT/e is the reduced temperature~we shall use,in the following, alsor* 5rs2), so as to calculate the equation of state of the fluid. The present estimates for the thmodynamical properties of the LJ fluid are in excelleagreement with previous simulation data31 ~see Tables I andII !, as well as with the values obtained using a semiempirequation of state.32 The excess entropy was computthrough the equation

s(ex)5b~u(ex)2 f (ex)!, ~5!

whereu(ex) and f (ex) are the excess energy and Helmhofree energy, respectively. A thermodynamic integration wperformed in order to computef (ex) , using a spline approximant for the data reported in Tables I and II:

b f (ex)~r!5E0

rdr8

r8FbP~r8!

r821G . ~6!


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The amount of orientational order that develops in the flfor increasing densities was assessed through the six-OCF:

h6~r !5^cos$6@u~r 8!2u~r 9!#%& , ~7!

wherer is the distance between two particles located atsitionsr 8 andr 9. The quantityu(r 8) is the angle formed by arandomly chosen nearest-neighbor bond atr 8 with a fixedreference axis. First neighbors of a particle in a given cfiguration were defined through the Voronoi construction.for the RDF, the histogram of the OCF was updated ev100– 500 MC cycles. A similar analysis was performedhard discs.

In order to complement our study of the RDF in 2D wisome information in 3D, we performed canonical NVT simlations of a 3D LJ system with 864 particles in two subcrical states, namely (r* 50.856; T* 50.75) and (r* 50.96;T* 51.15), and at the supercritical temperatureT* 52.74.We simulated a hard-sphere system as well at the redudensityr* 50.936.

IV. RESULTS

The RDFs of the LJ and HD fluids in 2D are shownFig. 1. The corresponding RMPE is plotted in Fig. 2 asfunction of the density. For later comparison with the Hsystem, we also show a few points obtained through a

TABLE I. Compressibility factor of a 2D LJ fluid forT* 50.7.

r*

bP/r

Present work Ref. 31

0.10 0.770.20 0.640.30 0.570.40 0.560.50 0.600.60 0.830.70 1.50 1.450.75 2.32 2.290.78 3.010.80 3.550.83 4.37 4.22

TABLE II. Compressibility factor of a 2D LJ fluid forT* 51.

r*

bP/r

Present work Ref. 31

0.10 0.920.20 0.880.30 0.90 0.890.40 0.97 0.960.50 1.150.60 1.540.70 2.43 2.390.75 3.24 3.190.80 4.42 4.390.83 5.330.856 6.19


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simulation of 2000 hard calottes embedded on the surfaca sphere.33,34All the curves displayed in Fig. 2 have a profithat is morphologically similar to that found in 3D: at lodensities,Ds is a negative, decreasing function ofr* until itreaches a minimum. Thenceforward,Ds starts growing rap-idly to, eventually, become positive. However, at significavariance with what has been already ascertained in relato the freezing of the LJ and HD models in 3D,2,3 the RMPEturns out to vanish at a densityr0* that is slightly but defi-nitely lower than the freezing-point density. In fact, atT*50.7 we findr0* .0.79 for the LJ system, while the curently estimated freezing density isr f* 50.83.31 In thehigher-temperature state (T* 51), r0* .0.82 andr f* 50.85,respectively.31 The RMPE of hard discs changes sign atr0*.0.83, while the freezing is expected to occur atr f* .0.88.26

Actually, the exact location of the freezing point and the venature of the transition in 2D are still controversial for bomodels. In fact, evidence of a narrow hexatic window, cloto the hitherto accepted freezing point, has been reporLarge-scale simulations of the 2D LJ system suggest theistence of a~possibly, metastable! hexatic state for tempera

FIG. 1. Radial distribution functions of the 2D models investigated: tLennard-Jones fluid at T* 50.7 and reduced densitiesr*50.70, 0.75, 0.79, 0.81; center, Lennard-Jones fluid atT* 51 and reduceddensitiesr* 50.75, 0.78, 0.82, 0.85; bottom, hard discs at reduced densr* 50.764, 0.802, 0.828, 0.853.


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tures around T* 52.16, at a fixed reduced pressuP* 520.29 The stability of such a phase has also been ptulated for hard discs, in the density interval 0.90– 0.9130

The thermodynamic behavior of the RMPE is consistent wthe hypothesis that the freezing of a fluid in 2D is anticipaby the growth of extended angular correlations. This stament can be made more precise by extracting the angcorrelation lengthj6(r) which characterizes the space bhavior of the OCF~see Fig. 3! at large distances. We fittethe heights of a few peaks ofh6(r ), assuming an exponentiadecay of the maxima as a function ofr. The fit included thefirst five peaks beyond the third shell, so as to avoidinterference of local-structure effects. The result of thanalysis is presented in Fig. 4. As expected, the angularrelation function increases steadily as a function of the dsity. However, both fluids exhibit two distinct correlatioregimes, as is clearly witnessed by the sharp change of sundergone byj6(r) in a very narrow range of densities. Ithe LJ fluid, the demarcation between the two regimesparticularly evident at low temperatures. As shown in Figat T* 50.7 the quantityj6(r) bends abruptly upward atdensity r* .0.79, which is also the threshold~marked inFig. 4 with an arrow! that was justindependentlyidentifiedthrough the contribution of multiparticlepositional correla-tions to the configurational entropy of the system. It thappears that the vanishing of the RMPE of the 2D LJ flurecords the onset of extended angular correlations whmay eventually be responsible for the emergence of a hexphase. This consistent, two-fold indication also persistshigher temperatures, but for a more rounded ‘‘knee’’j6(r) and its expected shift to larger densities.

Also in the HD case, the existence of two distinct rgimes is ostensibly manifested in the density dependencthe angular correlation length whose slope shows a ‘‘disc

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FIG. 2. Residual multiparticle entropy plotted as a function of the redudensity: full circles, 2D Lennard-Jones fluid atT* 50.7; full diamonds: 2DLennard-Jones fluid atT* 51; open circles: hard discs; squares: hard cottes~Ref. 33!. The lines are interpolating spline functions. An arrow idetifies the random sequential addition~RSA! threshold of hard discs~Ref.39!. The inset shows the excess~solid line! and pair entropy~dotted line! ofhard discs also plotted as a function of the reduced density. The crossundergone by the two curves corresponds to the zero-RMPE condition.density behavior of the above quantities in the LJ fluid is analogous toshown in the inset for hard discs.


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tinuity’’ at a density close to 0.86~see also Ref. 34!. How-ever, at variance with the LJ fluid, the RMPE of hard disvanishes at a density (r* .0.83) which slightly anticipatesthe sharp rise ofj6(r). Indeed, the buildup of extended agular correlations in the system is made possible by soprecursory structural phenomena. More specifically, we reto the pairing of disclinations to form dislocations and, cotextually, to the binding of dislocation pairs to form ‘‘rings,a process that is invoked by the KTHNY theory to explathe formation of the solid phase. An analysis of point defethat are observed in a dense fluid of hard calottes showsthe fraction of rings increases sharply atr* .0.83 at theexpense of isolated disclinations.35 This conclusion can alsobe safely extended to the HD fluid, since any ‘‘spurioueffect originated by the finite curvature of the host surfaemerges only at larger densities.34,35 This circumstance canalso be visually appreciated through an inspection of Figwhere it is apparent that the data referring to the RMPEhard calottes distribute smoothly along the curve relativethe HD fluid. The emergence of ordered structures wh‘‘prepare the stage’’ for the formation of angularly correlatpatterns is also reflected in the RDF of the HD fluid. In fajust atr* .0.83 we observe that the second maximum ofRDF starts deforming: its shape becomes asymmetrical ua shoulder appears for densities in the range correspondin

FIG. 3. Orientational correlation functions of the 2D models investigatethe same temperatures and densities as in Fig. 1.


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the position of the knee in the angular correlation lengSuch a shoulder eventually evolves into a distinct prepelocated at a characteristic distance of the triangular latcorresponding to the next-nearest-neighbor coordinashell, namelyr /a5A3, wherea5s(rCP/r)1/2 is the hex-agonal lattice constant on a plane andrCP is the HD densityat close packing. This feature had been explicitly observe2D by Prestipino Giarritta and co-workers,34 and has beendiscussed, more recently, in relation to the freezing of hacore systems.26 Of course, it is not surprising to verify thathe sprouting of orientational order is accompanied byappearance, on a local scale, of spatially ordered arraments that also become progressively resolved in the RAs to the nature of the information conveyed on this specmatter by the RMPE, the evidence discussed above cleshows that this structural indicator monitors the very figermination of order which, in a HD fluid, is heralded bclosely related precursory phenomena such as the formaof rings and of ‘‘minimal’’ clusters of hexagonally ordereparticles. We argue, retrospectively, that in a soft-copotential model~like the LJ fluid!, these phenomena show uconcurrently with the sharp rise of the angular correlat

tFIG. 4. Orientational correlation length plotted as a function of the redudensity: top, Lennard-Jones fluid atT* 50.7; center, Lennard-Jones fluid aT* 51; bottom: hard discs. An arrow marks the density where the residmultiparticle entropy is found to vanish.


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length at the same density threshold, sensitively recordethe vanishing of the RMPE.

We close this section with a comment on the positiormin , of the dip exhibited byDs(r). Giaquinta and Giuntadiffusely discepted, in their original article on this subjec2

on the significance of the minimum of the RMPE of haspheres, and also gave a critical overview of the relateddence on the thermodynamic, structural, and dynamicalhavior of the model.36 This evidence consistently leads to thconclusion that, for densities larger thanrmin , the conditionof the hard-sphere fluid is strongly reminiscent of that‘‘liquids’’ as originally defined by Bernal, namely ‘‘homo-geneous, coherent, andessentially irregularassemblages omolecules containing no crystalline regions or holes laenough to admit another molecule.’’37 In fact, beyond theminimum of the RMPE, such holes or cavities substantiadisappear.2 This condition, which at thermodynamic equilibrium applies only in a statistical sense, also has an intrigucounterpart in a different framework, such as that providby the random sequential addition~RSA! of nonoverlappingspheres onto a given volume. An RSA experiment is,definition, irreversible~particles are not allowed to move!and, thus, is nota priori significant for the properties of thsystem at thermodynamic equilibrium.38 In spite of this, Gi-aquinta and Giunta noted that the saturation density,rRSA, atwhich a system of hard spheres becomes jammed becauother particles can be added, coincides with the positionthe minimum inDs(r).2 In 2D, the RSA limit density is:rRSA.0.696.39 As seen in Fig. 2, this value correspondsthe position of the dip in the RMPE of discs. Summing upturns out that the change of behavior undergone byRMPE of hard spherical particles, from a decreasing toincreasing trend when plotted as a function ofr, occurs at adensity which corresponds to the RSA jamming limit both2D and in 3D. Such a characteristic threshold, that is mafested by the model at equilibrium as well as out of equilrium, marks the limit beyond which cooperative, i.e., intrisically many-body, effects come into play in determining tstate of the fluid.

V. THE SCALING OF THE RDF AND THEHANSEN–VERLET CRITERION

As discussed above, the RMPE of a simple 2D flusuch as HD or LJ particles, records—in a sensitive and rable way—the onset of extended orientational order insystem. As far as we can say, this indication is present away, viz., independently of the existence—in the phase dgram of the system—of a thermodynamically stable hexphase which may anticipate the freezing of the fluid. Tnature of this indication is consistent with the evidence gaered so far on other model systems in different thermonamic scenarios: in fact, as discussed in the introductionall cases investigated the vanishing of the RMPE pinpothe incipient ordering of the fluid into a more structurstate, not necessarily a crystalline solid. In this respect,zero-RMPE condition proves to be a rather general thoindirect ‘‘measure’’ of the degree of spatial order developin the system. In order to scrutinize this aspect further,compared the RDFs of both models for states wh


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Ds(r,T)50. However, for the comparison not to be biasby the ‘‘trivial’’ effect associated with the density, whosvalue changes slightly from one system to the other, wecaled the distances by referring them to theaveragesepara-tion between neighboring particles which is roughly proptional to r21/2. As is seen from Fig. 5, where we plotted thRDF of hard discs and of the LJ fluid at two supercritictemperatures, the three profiles do substantially coincaside from the first coordination shell whose detailed shobviously depends on the very nature of the interactiontential at short distances. Indeed, even for the LJ fluid,height of the first maximum is slightly different in the twstates that were singled out. The comparison unambiguoshows that, after scaling and as long asDs(r,T)50, theRDFs of the HD and LJ systems can be superimposed fthe first peak onwards. Such a scaling relation is not equeffective for the OCFs~see Fig. 6!, inasmuch as the amplitude of the oscillations is not the same for two correspondstates or systems, within the accuracy registered forRDFs (;1%). On thebasis of the evidence discussed abo

FIG. 5. Radial distribution functions of the 2D Lennard-Jones and hard-fluids in a state where the residual multiparticle entropy is found to vantriangles, Lennard-Jones fluid atT* 50.7 andr* 50.79; circles, Lennard-Jones fluid atT* 51 andr* 50.82; solid line: hard-disc fluid atr* 50.83.The contact value of the hard-disc RDF is 5.69.

FIG. 6. Orientational correlation functions of the 2D Lennard-Joneshard-disc fluids in a state where the residual multiparticle entropy is foto vanish: dotted line, Lennard-Jones fluid atT* 50.7 andr* 50.79; dashedline ~intermediate curve!, Lennard-Jones fluid atT* 51 andr* 50.82; solidline: hard-disc fluid atr* 50.83.


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we argue that the vanishing of the RMPE identifies a~struc-tural! condition of the fluid that is alike—at least, as farradial correlations are concerned—for both systems, inpendently of the specific thermodynamic state as well asthe detailed shape of the interaction potential.

Even in 3D, the RDFs scale, under the zero-RMPE cdition, in a very accurate way. Figure 7 shows the corsponding data for hard spheres as well as for the LJ fluidthree different thermodynamic states~one being supercriti-cal!. In this case, the interparticle distances were obvioureferred to the quantityr21/3. We recall that in 3D the vanishing of the RMPE is more directly related to the freezingthe fluid.2,3 This circumstance gives us the possibility of rconsidering the celebrated Hansen–Verlet~HV! freezingcriterion18 from a different perspective. The HV rule isone-phase criterion as is the currently investigated one.tually, the affinity between the two criteria becomes cleafter observing that the first maximum of the structure facresults, to a major extent, from the mapping in Fourier spof the long-range oscillations exhibited by the RDF in respace. More specifically, the frequency, damping, and amtude of these oscillations determine the position, width, aarea ~and, consequently, height! of the first maximum ofS(k).40 Hence, the HV criterion, upon fixing a ‘‘universal’reference value for the height of the first maximum of tstructure factor in order for freezing to occur, is actua‘‘tagging’’ a related amplitude of the oscillations observedthe RDF profile. But, the existence of a characteristic spaprofile associated with the incipient ordering of the fluidprecisely what follows from the RMPE criterion through thscaling of the RDFs in those thermodynamic states whereconditionDs(r,T)50 is satisfied. We conclude that the twcriteria manifestly rest on a common rationale, namelyidentification of a specific condition of the fluid which,aposteriori, unravels as the underlying stage for the occrence, at a macroscopic level, of a thermodynamic transiinto a more ordered state. Note, however, that, at variawith the ad hocformulation of the HV rule, which rests onan external input~i.e., the height of the principal maximum

FIG. 7. Radial distribution functions of the 3D Lennard-Jones and hasphere fluids in a state where the residual multiparticle entropy is founvanish: squares, Lennard-Jones system atT* 50.75 andr* 50.86; triangles,Lennard-Jones system atT* 51.15 andr* 50.96; circles, Lennard-Jonesystem atT* 52.74 and r* 51.11; solid line: hard-sphere fluid atr*50.94. The contact value of the hard-sphere RDF is 5.65.


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of the structure factor at the transition point!, the vanishingof the RMPE is an internal, self-contained statement whphysical content is rather straightforward, in view of the fathat it more directly hits at the very core of the phenomenogy that is being investigated. Last, as already recalled inintroduction, the zero-RMPE ordering criterion is more geeral than the HV freezing rule inasmuch as it applieswithoutmodificationsto a larger variety of physical situations both2D and 3D, not just freezing.

VI. SUMMARY AND CONCLUDING REMARKS

In this article we have revisited the structural propertof simple dense fluids in two dimensions~2D!, in the frame-work provided by the multiparticle correlation expansionthe configurational entropy.1 In particular, we computed—using Monte Carlo simulation techniques—a quantity whweighs up the contribution of many-body~where ‘‘many’’stands for triplets, at least! correlations in the overall balancof states that are available to the system at given densitytemperature. Such a quantity, called residual multipartentropy~RMPE!, is a sensitive indicator of the ordering phnomena which take place in a fluid in rather disparate thmodynamic conditions. In the case of hard discs aLennard-Jones particles, the vanishing of the RMPE wnesses an accelerated growth of bond-angle correlatwhich preludes to the freezing of the fluid into a solid phaThe search for structural precursors to the freezing of simfluids has brought many authors to introduce one-phaseteria, which aim at locating the transition without resortingthe comparison of the free energies of the coexisting phaHowever, aside from other conceptual aspects, most of sempirical rules fail, in their original 3D formulation, in alower dimensionality. In this respect, the zero-RMPE contion, used as a preliminary ordering criterion in relation tovariety of thermodynamic phenomena, keeps valid in 3Dwell as in 2D.

Finally, the comparison of the long-range profiles of tradial distribution function in those states where the RMis zero for each of the two systems, suggests an illuminacorrespondence with the freezing criterion formerly intrduced by Hansen and Verlet.

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entropy, correlations, and ordering in two dimensions

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