Hyper-parallel tempering Monte Carlo: Application to the Lennard-Jones fluid and the restricted primitive model

Hyper-parallel tempering Monte Carlo: Application to the Lennard-Jonesfluid and the restricted primitive modelQiliang Yan and Juan J. de Pablo Citation: J. Chem. Phys. 111, 9509 (1999); doi: 10.1063/1.480282 View online: http://dx.doi.org/10.1063/1.480282 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v111/i21 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 21 1 DECEMBER 1999

Hyper-parallel tempering Monte Carlo: Application to the Lennard-Jonesfluid and the restricted primitive model

Qiliang Yan and Juan J. de PabloDepartment of Chemical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706

~Received 14 July 1999; accepted 7 September 1999!

A new generalized hyper-parallel tempering Monte Carlo simulation method is presented. Themethod is particularly useful for simulation of many-molecule complex systems, where roughenergy landscapes and inherently long characteristic relaxation times can pose formidable obstaclesto effective sampling of relevant regions of configuration space. In this paper, we demonstrate theeffectiveness of the new method by implementing it in a grand canonical ensemble for theLennard-Jones fluid and the restricted primitive model. Coexistence curves and critical behaviorhave been explored by the new method. Our numerical results indicate that the new algorithm canbe orders of magnitude more efficient than previously available techniques. ©1999 AmericanInstitute of Physics.@S0021-9606~99!50945-8#

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I. INTRODUCTION

Molecular simulations of complex systems are difficuparticularly at low temperatures, because configurationsget easily trapped in local energy minima, thereby precludsampling of other, relevant regions of phase space. Theherently long characteristic relaxation times that often pvail in complex fluids further aggravate matters. Over tlast few years several, powerful methods have been propto overcome the first difficulty; some examples are providby multicanonical sampling,1,2 1/k sampling,3 simpletempering,4,5 expanded ensembles,6 J-walking,7,8 and paralleltempering.9–11 While these methods are relatively effectivat overcoming high-energy barriers, they do little to ‘‘accerate’’ the slow relaxation of complex, many-molecule sytems at low temperatures.

It has been increasingly recognized that open ensemprovide an effective means for overcoming slow-relaxatproblems; molecules can get in and out of a system, thercircumventing diffusional bottlenecks. In this paper we drelements from simple and parallel tempering, J-walking,panded ensembles, and histogram reweighting to proponew, powerful Monte Carlo method for simulation of manmolecule systems. In doing so we combine the proven befits of tempering or extended ensemble techniques wthose of open-ensemble simulations. In this new methconfigurations can hop simultaneously along several insive variables. Furthermore, our formulation is well-suitfor further combinations of expanded ensemble~or ‘‘simpletempering’’! with parallel tempering. To distinguish oumore general formulation from conventional~one-variable!parallel tempering, throughout this manuscript we referthe new method as ‘‘hyper-parallel tempering Monte Carl~HPTMC!. As shown in this manuscript, hyper-parallel tempering is significantly more efficient than existing methofor simulation of phase transitions. The new algorithmrelatively simple, and has the added benefit of being eaincorporated into molecular-dynamics or Monte Carlo simlation programs with minor modifications to existing code

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In order to demonstrate the usefulness of HPTMC,this paper, we have chosen to work with two systems: TLennard-Jones fluid and the restricted primitive modelelectrolyte solutions. On the one hand, the Lennard-Jofluid has generally been adopted as a benchmark systemwhich to test new algorithms. Extensive studies of its phbehavior facilitate significantly comparisons to literature dand to existing algorithms. On the other hand, the restricprimitive model has attracted considerable interest overpast decade; its liquid–liquid phase transition at low teperatures has posed a challenge to theoreticians, and liture data have often been called into question. WeHPTMC to study that phase transition.

The paper is organized as follows. We begin by descing in detail how hyper-parallel tempering works. We thdefine the models and give a brief description of methological technicalities used in our work, such as histogrreweighting and finite-size scaling techniques. We thpresent the results for the Lennard-Jones fluid and thestricted primitive model, and we compare these to availadata. We conclude the paper by discussing the advantand disadvantages of the new method, and by outlining soof its possible future applications.

II. HYPER-PARALLEL TEMPERING MONTE CARLO

For the sake of generality, we consider an arbitrarysemble whose partition function is given by

Z5(x

V~x!w~x!)j

exp~ f j qj~x!!, ~1!

wherex denotes the state of the system,V(x) is the densityof states,w(x) is an arbitrary weighting function for statex,f j ’s are generalized forces or potentials, and theqj ’s are thecorresponding conjugate generalized coordinates of thetem. A grand canonical ensemble can thus be recoveresetting

w~x!5exp~2bU~x!!, f 5bm, q~x!5N~x!, ~2!

9 © 1999 American Institute of Physics

icense or copyright; see http://jcp.aip.org/about/rights_and_permissions

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9510 J. Chem. Phys., Vol. 111, No. 21, 1 December 1999 Q. Yan and J. J. de Pablo

whereb51/kBT, T is temperature,U(x) is the potential en-ergy corresponding to configurationx, m is the specifiedchemical potential, andN(x) is the number of particles inconfigurationx. Alternatively, an isobaric-isothermalNPTensemble can be recovered by setting

w~x!5exp~2bU~x!!, f 52bP, q~x!5V~x!, ~3!

whereV(x) is the volume that corresponds to configuratix.

We now consider a composite ensemble consisting oM,noninteracting replicas of the above mentioned ensemeach at a different set of generalized forces and weighfunction. The complete state of the composite ensemblspecified throughx5(x1 , x2 , . . . ,xM)T, wherexi denotesthe state of theith replica. We define the partition function othe composite ensemble according to

Zc5)i 51

M

Zi . ~4!

The un-normalized probability density of the composite stx is given by

p~x!5)i 51

M

wi~xi !)j

exp~ f j ,i qj~xi !!. ~5!

To sample configurations from the composite ensema Markov chain is constructed in such a way as to asymtotically generate configurations according to the limitidistribution function appearing in Eq.~5!. Two types of trialmoves are used to realize that Markov chain:

~1! Standard Monte Carlo trial moves are used to locaupdate each of the replicas of the system. Since repldo not interact with each other, standard Metropoacceptance–rejection criteria~for the underlying en-semble! are employed within each replica.

~2! Configuration swaps are proposed between pairs oflicas i and i 11, so that

xinew5xi 11

old ,~6!

xi 11new5xi

old .

To enforce a detailed-balance condition, the pair of replito be swapped is selected at random, and the trial swaaccepted with probability:

pacc~xi↔xi 11!5minF1,wi~xi 11! wi 11~xi !

wi~xi ! wi 11~xi 11!

3)j

exp~2D f j Dqj !G , ~7!

whereD f j5 f j ,i 112 f j ,i is the difference in generalized forcf j between the two replicas, andDqj5qj (xi 11)2qj (xi) isthe difference of the corresponding conjugate generalizedordinates.

So far tempering has been discussed for the caswhich several fields are sampled at the same time. A furgeneralization is to now couple the technique to that ofpanded ensembles~or ‘‘simple tempering’’!, so one can


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simulate long polymer chains at high densities more eciently. In this latter generalization, the whole simulatiosystem consists of several expanded grand canonicalsembles. Each replica includes a tagged chain, whose lefluctuates during the simulation. When a swap trial moveattempted, these tagged chains are also switched. The csponding acceptance criterion can be obtained by followindevelopment analogous to the one presented above. Mdetails of the generalization and its application are givelsewhere.27

III. MODELS AND SIMULATION METHODS

A. Models

In this work, we apply the hyper-parallel temperinmethod to two different molecular models. The first is tLennard-Jones fluid. The potential energy between a paiparticle i and j, separated by a distancer i j , is defined by

Ui j 54eF S s

r i jD 12

2S s

r i jD 6G , ~8!

wheree is the depth of the attractive well, ands is a param-eter that controls the size of the particles. In this work,Lennard-Jones potential-energy function is truncated acutoff distancer c . To be consistent with previous studiewe chooser c52.5s, and the potential is left unshifted.

The restricted primitive model~RPM! of an ionic solu-tion consists of 2N hard spheres of diameters, half of themcarrying a negative charge and the other half carrying a ptive charge. A solvent is not modeled explicitly; instead, itsimply considered as a dielectric continuum with dielectconstantD. The charged spheres interact via a Coulombtential,

Ui j 5H 1`, r i j >s

e2

4pDD0

zizj

r i j, r i j ,s

, ~9!

wherezie and zje are the charges carried by ionsi and j,respectively, e is the charge of the electrone51.602310219 C, andD0 is the dielectric permeability of vacuumD058.85310212 C2 N21 m22. Note that the restrictedprimitive model requires thatuzi u5uzj u.

Throughout this work results are reported in reducunits. For the Lennard-Jones fluid, the reduced temperais defined asT* 5kBT/e, wherekB is Boltzmann’s constantThe reduced density is defined asr* 5Ns3/V, with N beingthe number of particles andV the volume of the simulationbox. For the restricted primitive model, the reduced tempeture is defined asT* 54pDD0skBT/e2, and the reduceddensity is given byr* 52Ns3/V.

B. Simulation details

In this work, the hyper-parallel tempering methodimplemented in the grand canonical ensemble. By substing Eq. ~2! into Eq. ~7!, we arrive at the following acceptance criteria for swapping two replicas:

pacc~xi↔xi 11!5min@1,exp~DbDU2D~bm!DN!#~10!


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9511J. Chem. Phys., Vol. 111, No. 21, 1 December 1999 Hyper-parallel tempering Monte Carlo

where Db5b i 112b i , DU5U(xi 11)2U(xi), D(bm)5b i 11m i 112b im i , andDN5N(xi 11)2N(xi).

Two kinds of trial moves are employed, namely, trdisplacements of particles and trial creations or destructof particles. The relative frequency of these two kinds of trmoves is set to be unity. For the restricted primitive modthe conventional Ewald sum method is used to calculong-range contributions to the energy arising from the slodecaying Coulomb potential. Conducting boundary contions are employed in our calculations; It has been poinout in the literature that such a boundary condition is esstial for simulations of ionic systems.20,21 Particle creationsand destructions are conducted in pairs to preserve eleneutrality. A distance-bias Monte Carlo scheme is also eployed to further facilitate particle transfer moves.23

During the simulation, the number of particlesN and thetotal potential-energyU of each replica is recorded. The joindistributionsp(N,U) are accumulated in the form of a histogram. Note that for the restricted primitive model,N is thenumber of particle pairs~the chemical potential is defined athe total chemical potential per ion pair!.

C. Histogram reweighting technique

The histogram reweighting technique12–14 is designed toextract a maximal amount of information from the resultsa set of molecular simulations. It has been widely useddetermine the near-critical and sub-critical coexistence prerties of fluids. For completeness, in this work we providbrief description of its implementation; for additional detaireaders are referred to the original publications. In a grcanonical simulation, the relevant thermodynamic variabare the temperatureT and the chemical potentialm. A simplegrand canonical simulation can be conducted atT5T0 andm5m0, and a two-dimensional histogram of energy anumber of molecules can be constructed. The entries to sa histogramH(N,U) represent the number of times that tsystem is observed withN particles and potential-energyU.If the total number of realizations of the system is denotedK, the probabilityP(N,U,T0 ,m0) that the system hasN par-ticles and energyU at T0 andm0 is given by

P~N,U,T0 ,m0!5H~N,U !/K . ~11!

The probability distribution for a grand canonical ensemis given by

P~N,U,T,m!5V~N,V,U !exp~2bU1Nbm!

J~m,V,T!, ~12!

where V(N,V,U) is the microcanonical partition functio~density of states atN andU), andJ is the grand partitionfunction, given by

J~m,V,T!5(N

(U

V~N,V,U !exp~2bU1Nbm!.

~13!

Combination of Eqs.~11! and ~12! provides a MonteCarlo estimate ofV(N,V,U), given by

V~N,V,U !5wH~N,U !exp~b0U2Nb0m0!, ~14!


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where w5J(m,V,T)/K is a proportionality constant. BecauseV(N,V,U) is independent ofT andm, the probabilitythat the system hasN particles and energyU at a differentstate pointT andm can be estimated according to

P~N,U,T,m!

5H~N,U !exp@2~b2b0!U1N~bm2b0m0!#

(N(UH~N,U !exp@2~b2b0!U1N~bm2b0m0!#.

~15!

The average value of any function ofN andU can thereforebe estimated from

^A&5(N

(U

A~N,U !P~N,U,T,m!. ~16!

In practice, the extrapolation scheme provided by Eq. 15only useful if the distance between (T0 ,mp) and (T,m) is nottoo large. The range of energy and number of particles cered by a single histogram is limited, and extrapolationsyond that range are unfounded.

In order to expand the applicability of the histogram rweighting technique, several simulations can be carriedat different state points. Such simulations have traditionabeen conducted independently of each other, i.e., in seriethis work, such simulations are conducted in parallel accoing to our tempering scheme; each replica correspondsdifferent state point, and the resulting histograms are cobined to generate thermodynamic predictions over a wrange of conditions.

The underlying bridge to combine multiple histogramsthe fact that the microcanonical partition functioV(N,V,U) is independent of temperature and chemical ptential. Estimates ofV(N,V,U) by Eq. ~14! from differentsimulations should be consistent with each other~within thestatistical uncertainty of the calculations!. If histograms fromdifferent runs overlap sufficiently, it is possible to find a sof proportionality constants$w% so that this requirement isfulfilled. An optimal estimate ofV(N,V,U) can then be de-termined as a weighted average ofV ’s extracted from dif-ferent runs. Ferrenberg and Swendsen proposed an effimethod to determine the sought-after proportionality costants and the weighting factors: forR simulation runs, theoptimal ~unnormalized! probability distribution is given by

P~N,U;m,b!5(n51

R Hn~N,U !exp~2bU1Nbm!

(m51R Km exp~2bmU1Nbmmm2Cm!

,

~17!

whereKm is the total number of realizations in runm. TheconstantsCm must be determined self-consistently by titerative relationship

exp~Cm!5(N

(U

P~N,U;mm ,bm!. ~18!

In this work, we use a quasi-Newton method to solvenonlinear equations@Eqs.~17! and ~18!#.

At sub-critical temperatures, the grand canonical dendistribution is characterized by a double peaked structuprovided the chemical potential is close to the coexiste


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value. If the so-called ‘‘two-state’’ approximation iadopted,19 the criteria for phase equilibrium are equivalentrequiring that the areas under the two peaks be equal.determination of the precise location of a coexistence pcan, therefore, be achieved by simply tuning the chempotential at any given temperature, until the areas undertwo peaks become the same; the coexisting densities otwo coexisting phases then correspond to the mean densunder these two peaks.

D. Mixed-field finite-size scaling

The periodic boundary conditions generally employedmolecular simulations suppress long-range fluctuations; na critical point, thermodynamic properties calculatedsimulation are, therefore, different from those correspondto the thermodynamic limit. However, it is possible to esmate with high accuracy the coordinates of the critical poby resorting to finite-size scaling techniques.15,16

For the systems studied here, the relevant scaling ficonsist of combinations of the temperature and chemicaltential and are given by

t5bc2b1s~m2mc!, ~19!

h5m2mc1r ~bc2b!,

wheret is the thermal scaling field,h is the ordering scalingfield, and subscriptc serves to denote a quantity evaluatedthe critical point. Parameterss and r, which control the de-gree or extent of field mixing, are system specific.

Conjugate to the two scaling fields are two scaling oerators, namely, the ordering operatorM and an energy-likeoperatorE. They consist of linear combinations of the paticle densityr and the energy densityu5U/V

M51

12sr~r2su!, ~20!

E51

12sr~u2rr!.

For the simple case of models with Ising symmetry~wheres5r 50), M is simply the magnetization andE is just theenergy density.

In the critical region, the probability distributions ofMandE exhibit a scaling behavior of the form

PL~M!5ALb/nPM* ~ALb/ndM!,~21!

PL~E!5BL(12a)/nPE* ~BL(12a)/ndE!

wherea, b, andn are universal critical point exponents. thfixed point limiting operator distributionsPM* and PE* arealso universal for all systems of a given universality claThe quantitiesdM anddE measure deviations from criticaity; dM5M2Mc , dE5E2Ec ; A and B are system-specific constants.

For the Ising universality class, the universal distributifunctions mentioned above can be estimated from hiresolution Monte Carlo simulations.17 To estimate the criticapoint of an Ising-class fluid, the temperature, chemicaltential and the mixing parameters are tuned so that the


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sulting distributionsPL(M) and PL(E) collapse onto thoseof the Ising model. Values corresponding to the thermonamic limit can be estimated by extrapolating finite-size vues according to the scaling behavior

Tc~L !2Tc~`!}L2(u11)/n, ~22!

rc~L !2rc~`!}L2(12a)/n, ~23!

where a, u, and n are universal exponents. For the 3~three-dimensional! Ising universality class,a'0.11, u'0.54,b'0.312, andn'0.629.

IV. RESULTS AND DISCUSSION

A. Results for Lennard-Jones fluids

In our calculations for Lennard-Jones fluids, we usedreplicas, each having a box sizeL57s. The temperature andchemical potential of each replica are reported in TableThese values were selected to guarantee frequent swaptween replicas. Configuration swaps were attempted ev10 Monte Carlo steps; a total of 106 Monte Carlo steps wereperformed to generate the energy and density histogramquired to construct joint (N,U) distributions for each replica

During a simulation, we keep track of ‘‘physical’’ replicas as well as ‘‘logical’’ replicas. A physical replica is aactual collection of atoms that we follow throughout thcourse of the simulation. A logical replica is whatever cofiguration happens to visit a specific box at some speciconditions of temperature and chemical potential~e.g., boxi,at m i and Ti). Figure 1 shows the evolution of a logicareplica atT* 50.73 andbm525.30, as a function of MonteCarlo steps; the ordinate axis indicates which physical rlica happens to be visiting the logical replica atT* 50.73 atany given time during the simulation. This serves to illustrahow configurations are swapped during the course ofsimulation. As can be inferred from Fig. 1, a physical replivisits each logical replica relatively frequently and unformly. After a successful configuration swap, two logicreplicas adopt a completely new configuration. Further, cfigurations corresponding to logical replicas at low tempetures and high densities are ‘‘passed’’ over to logical replicat higher temperatures and lower densities; such configtions can then relax more expeditiously and then be pasback to lower temperature replicas. This process greatly

TABLE I. Temperatures and chemical potentials used for the simulationthe truncated Lennard-Jones fluid, whereT* 5kBT/«, with kB being Boltz-mann’s constant.

Replica T* bm Replica T* bm

1 1.20 22.74 10 0.82 24.522 1.17 22.83 11 0.79 24.763 1.11 23.02 12 0.76 25.024 1.07 23.18 13 0.73 25.305 1.03 23.35 14 0.70 25.636 0.99 23.51 15 0.67 25.997 0.94 23.75 16 0.64 26.388 0.90 23.98 17 0.62 26.669 0.86 24.23 18 0.60 26.97


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celerates the relaxation of the global system and facilitasampling of phase space under adverse conditions ofperature and chemical potential.

Figure 2 shows the probability density of the margindistribution for the number of particles in the logical repliat T* 50.73 andbm525.30. The chemical potential of thareplica corresponds to a thermodynamic state slightly offsaturated liquid line. We would like to emphasize the occrence of two distinct peaks in that density distribution. Givthe fact that the temperature is well below the critical teperature, the observed ‘‘tunneling’’ behavior between a cdensed phase and its vapor indicates that the largeenergy barrier associated with the vapor–liquid phtransition can be overcome by the hyper-parallel tempemethod. Such a crossing of high-energy barriers greatlycilitates the simulation of phase transitions. In contrast tmulticanonical sampling method, this tunneling is achievby simple configuration swaps, rather than by artificial, triand-error flattening of free energy barriers; intermedisamples in unstable regimes are not necessary in this arithm.

To demonstrate how accurately the new methodsample the relative weights of the two peaks, we also sh

FIG. 1. Replica number as a function of Monte Carlo steps, forT* 50.73andbm525.30.

FIG. 2. Probability density of the marginal distribution of number of pticles for T* 50.73, bm525.30. The solid line represents the original hitogram for the system at these conditions. The dashed line depicts thebined histogram that results from a full histogram reweighting analysis oresults at all conditions studied here~i.e., it is the ‘‘true’’ histogram!.


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in the figure the ‘‘real’’ density probability distribution corresponding the same thermodynamic conditions; the lawas calculated from histogram reweighting of data obtainfrom all 18 boxes. For clarity, that distribution is shifted b0.05.

Figure 3 shows the phase diagram calculated fromtograms corresponding to all 18 logical replicas. Also shoare literature data for the same fluid.18 As expected from acorrect algorithm, the agreement between the two sets ofis satisfactory. The slight discrepancies at high temperatuare due to different definitions of the equilibrium saturatdensity. In this work we regard the mean density correspoing to a peak of the distribution as the equilibrium valuWilding defines it as the peak value of the distribution. Acan be seen in Fig. 3, the proposed method is able to geate phase equilibrium data at temperatures and densitiethe near vicinity of the triple point of the truncated LennarJones fluid~e.g.,T* 50.60 andr* 50.86). A simple Gibbsensemble method or conventional grand canonical simtions would be difficult and unreasonably demanding unsuch conditions.

FIG. 4. Autocorrelation function for the average potential energy per afor a Lennard-Jones fluid, corresponding to the lowest temperature re(T* 50.60, bm526.97). The solid line corresponds to results of hypeparallel tempering simulations. The dashed line shows results from aventional grand canonical Monte Carlo simulation.

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FIG. 3. Phase diagram~vapor–liquid equilibria! for a truncated Lennard-Jones fluid. The squares correspond the results of this work, and theangles show results reported by Wilding~Ref. 18!. Statistical errors aresmaller than the symbol size. The solid line is an Ising form fit to tsimulation data.


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As a measure of the efficiency of the new method,show in Fig. 4 the autocorrelation function of the averapotential energy per atom, evaluated at the lowest tempture replica studied here (T* 50.60 andr* 50.86). As canbe seen in the figure, for hyper-parallel tempering the eneautocorrelation function decays to zero after about 20Monte Carlo~MC! steps. For a conventional grand canonicsimulation, it decays to zero after about 20 000 steps. Fan energy autocorrelation point of view, hyper-parallel tepering is about one order of magnitude more efficient ththe conventional method. Note, however, that for simulatioof phase transitions, a more appropriate measure ofciency is provided by the ‘‘tunneling time,’’ i.e., the timrequired to observe a jump from a vapor-like phase toliquidlike phase during a simulation. A conventional methis unable to produce such jumps, at least not in a reasonamount of time; hyper-parallel tempering routinely gives rto such jumps. In this regard, the new method is consideramore efficient than existing algorithms for open ensembDifficult simulations that were not possible with previousavailable techniques could become feasible with HPTMC

B. Results for the restricted primitive model

In this work, four different box sizes are used for threstricted primitive model:L513, 15, 16, and 17s. Thelength of the runs is about 1 to 23106 MC steps, depending

FIG. 5. Order parameter distributions for the restricted primitive modeltheir matching to the limiting Ising universal form. Circles:L513s;Squares:L516s; Diamonds:L517s.

TABLE II. Temperatures and chemical potentials used for the simulatiothe restricted primitive model (L513s), where T* 54pDD0skBT/e2,whereD is the dielectric constant,D0 the dielectric permeability of vacuumkB Boltzmann’s constant, ande is the charge of the electron.

Replica T* bm Replica T* bm

1 0.052 225.82 8 0.045 229.602 0.050 226.79 9 0.044 230.253 0.0492 227.19 10 0.043 230.924 0.0485 227.55 11 0.042 231.665 0.0475 228.10 12 0.041 232.436 0.0468 228.50 13 0.040 233.247 0.046 228.98


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on the box size. For theL513s box, 13 replicas are used tget the coexistence curve. For other box sizes, 3–6 replare used, mainly to explore the behavior of the model nthe critical point. The coexistence curve near the critipoint, of course, can also be obtained by these runs. Agthe values of temperature and chemical potential of threplicas are chosen close enough to each other so thatfiguration swaps can occur frequently. Also, they are choto be in the neighborhood of the coexistence curve. Tablreports the values forL513s.

We estimated the critical point of the restricted primitivmodel by matching the order parameter distribution ontoIsing universal form.17 Figure 5 shows the result of sucmatching. As the runs are not too long~only 1 to 23106 MCsteps!, the quality of the matching is not perfect, but it is stsatisfactory. We also observed that for the smaller boxthere are not enough data points on the low-density side;the largest box size, however, the situation is much betThe box-size dependence of the critical temperaturecritical density is shown in Figs. 6 and 7, respectively. Oestimates of critical temperature and density for infinite b

d

FIG. 6. Critical temperature scaling with system size for the restricprimitive model. The circles show results of this work, and the squares sdata reported by Orkoulaset al. ~1999!; note that error bars were not reported by these authors.

FIG. 7. Critical density scaling with system size for the restricted primitmodel. The circles show results of this work, and the squares showreported by Orkoulaset al. ~1999!; note that error bars were not reported bthese authors.

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size, along with those reported by earlier researchers,summarized in Table III. Within the uncertainty of the caculations, our results are consistent with those reportedearlier researchers.23,24 Note, however, that our simulationand extrapolations to infinite size suggest that the prevalue of rc* is slightly lower than that reported in previoustudies. In addition, Figs. 6 and 7 also show that the slopethe linear relationship between the critical parameterssystem size are quite different from those reported by Ooulaset al.23

The coexistence curve of the restricted primitive mois shown in Fig. 8. Our results are compared with earlier dreported by Panagiotopoulos,25 Caillol,26 and Orkoulas andPanagiotopoulos.22,23Our results exhibit small but systematdeviations from the latest data reported by OrkoulasPanagiotopoulos.23 On the high-density side, our results asystematically smaller than the reported data, while onlow-density side, our results are slightly higher. At this poithe origin of such disagreement is uncertain. One possorigin could be that in our simulations, metal~conducting!boundary conditions are applied in the calculation ofEwald sum, while in Orkoulas and Panagiotopoulos wovacuum boundary conditions were employed. As mentioearlier, it has been argued in the literature that conducboundary conditions should be used for simulationscharged systems. The source of the discrepancy could alsthat the simulation method employed here is more efficithan that of Orkoulas and Panagiotopoulos, and it therefpermits simulations of larger systems with a smaller degof correlation between successive configurations; our res

FIG. 8. Coexistence curve for the restricted primitive model. The filcircles show results of this work, the open circles show results reporteOrkoulas and Panagiotopoulos~1999!, the squares are results reportedPanagiotopoulos~1992!, the triangles show results reported by Caill~1994!, and the diamonds are results reported by Orkoulas and Panagpoulos~1994!.

TABLE III. Critical point parameters in the limit of infinite system size.

Author Tc* rc*

This work 0.049260.0003 0.06260.005Orkoulaset al. ~1999! 0.049060.0003 0.07060.005Caillol et al. ~1997! 0.048860.0002 0.08060.005


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correspond to systems that are generally much largerthose employed by Orkoulas and Panagiotopoulos.23

In that regard, we would like to draw special attentionthe computational effort required for our simulations. Amentioned above, 106 configurations were used to construthe coexistence curve. Multiplying by the number of boxestotal of about 107 configurations were generated. Orkouland Panagiotopoulos report that, to obtain results of comrable accuracy using a grand canonical method, about9

configurations are necessary. These numbers suggest thnew hyper-parallel tempering method is about two ordersmagnitude faster than grand canonical ensemble simulatof comparable systems.

V. CONCLUSIONS

In this work, we have presented a new generalizhyper-parallel tempering Monte Carlo simulation methoThe method has been shown to be superior to conventisimulation techniques for the study of phase transitionsfluids. The method has the added benefit of being remarkasimple to implement. While some preliminary trial and errruns is necessary to optimize the performance, we foundthis could be done with relative ease. We demonstrated sof its advantages by applying it to the simulation of a simptruncated Lennard-Jones fluid, down to temperatures closthe triple point, and to the simulation of the restricted primtive model. Our results for the Lennard-Jones fluid are copletely consistent with the literature. Our results for the PRexhibit small but systematic deviations from those reporby previous authors for smaller systems. Our results ashow that the new method is capable of overcoming the lafree energy barrier associated with a vapor–liquid or liquiliquid phase transition, and that it is also capable of relaxthe system much faster than traditional grand canonicalsemble simulations.

One disadvantage of the method is that a good setemperatures and chemical potentials must be chosen inder to maximize the efficiency of the method. This canachieved by a few short test runs in small systems. Tmajor considerations must be taken into account for selecof appropriate simulation conditions:

~1! For simulation of coexistence, the state points shouldnear the coexistence curve. For lower temperatures,better to be slightly off to the liquid side; since the vapdensity is very small at low temperatures, simulationssuch low densities usually do not provide much informtion.

~2! The state points of neighboring replicas must be closeeach other, so that enough configuration swaps cancur. As the box size becomes large, state points muscloser; this is a reflection of the fact that the relevadistribution functions become narrower as the systgets larger.

The first consideration is not unique to HPTMC, butoccurs whenevermVT and histogram reweighting techniqueare employed. Fortunately, the chemical potential can benipulated with relative ease, particularly for small system

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to explore the phase behavior approximately. Our experiesuggests that this preliminary search of phase space reqonly a small fraction of the overall computer time requirments.

The second consideration also arises whenever multtogram reweighting techniques are employed to analyzefrom a series of simulations. Unless the degree of ovebetween histograms is sufficiently large, multihistogramweighting techniques are not useful for calculating phdiagrams. By construction, if two histograms overlap suciently, the HPTMC method gives rise to a reasonableceptance rate for swap trial moves. One could thereforegue that if a histogram reweighting technique is to be usHPTMC does not incur in additional computational requiments and it improves efficiency significantly. In this worour selection of state points led to an acceptance rate;20%–30% for configuration swap attempts. Note, hoever, that at these preliminary stages of our work we hnot made any attempts to optimize the swapping rateexamine its effects on overall efficiency.

ACKNOWLEDGMENTS

This work was supported through a PECASE awafrom the National Science Foundation to J.J.dP and throaward CTS-9901430. J.J.dP. is also grateful to the DreyFoundation for a Teacher-Scholar Award. J.J.dP. wisheacknowledge the hospitality extended to him by GustaChapela during a sabbatical stay at the Mexican PetroleInstitute.


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Hyper-parallel tempering Monte Carlo: Application to the Lennard-Jones fluid and the restricted primitive model