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Theoretical foundations of dynamical Monte Carlo simulations

Kristen A. Fichthorn

Department

of

Chemical Engineering, Pennsylvania State University, University Park, Pennsyivania 16802

W. H. Weinberg

Department

of

Chemical Engineering, University

of

California, Santa Barbara, California 93106

(Received 12 December 1990;accepted10 April 1991)

Monte Carlo methods are utilized as computational tools in many areasof chemical physics. In

this paper, we present he theoretical basis or a dynamical Monte Carlo method in terms of

the theory of Poissonprocesses.We show that if: ( 1) a “dynamical hierarchy” of transition

probabilities s createdwhich also satisfy the detailed-balance riterion; (2) time increments

upon successfuleventsare calculated appropriately; and (3) the effective ndependence f

various eventscomprising the systemcan be achieved, hen Monte Carlo methods may be

utilized to simulate the Poissonprocessand both static and dynamic propertiesof model

Hamiltonian systemsmay b e obtained and interpreted consistently.

1. INTRODUCTION

Monte Carlo methods are utilized as computational

tools in ma ny areas of chemical physics.‘*’ Although this

techniquehas been argely associatedwith obtaining static,

or equilibrium properties of model systems, Monte Carlo

methodsmay also be utilized to study dynamical phenome-

na. Often, the dynamics and cooperativity leading o certain

structural or configurational properties of matter are not

completely amenable o a macroscopiccontinuum descrip-

tion. On the other hand, molecular dynamics simulations

describing he trajectories of individual atoms or molecules

on potential energy hypersurfacesare not computationally

capableof probing large systemsof interacting particles at

long times. Thus, in a dynamical capacity, Monte Carlo

methodsare capableof bri dging the ostensibly a rge gap ex-

isting between these two well-established dynamical ap-

proaches, ince he “dynamics” of individal atomsand mole-

cules are modeled in this technique, but only in a

coarse-grainedway representing average features which

would arise rom a lower-level result.

The application of the Monte Carlo method to the study

of dynamical phenomena equiresa self-consistentdynami-

cal interpretation of the techniqueand a set of criteria under

which this interpretation may be practically extended. n

recent publications,3t4

certain inconsistencies have be en

identified which arise when the dynamical interpretation of

the Monte Carlo method is loosely applied. These studies

have emphasized hat, unlike static properties, which must

be identical for systems having identical model Hamilto-

nians, dynamical properties are sensitive o the manner in

which the time seriesof eventscharacterizing the evolution

of a system s constructed. In particular, Monte Carlo stud-

ies comparing dynamical properties simulated away from

thermal equilibrium have revealeddifferencesamong var-

ious sampling algorithms.3” These studies have under-

scored he importance of utilizing a Monte Carlo sampling

procedure n which transition probabilities are basedon a

reasonable ynamical model of a particular physical phe-

nomenonunder consideration, n addition to satisfying the

usual criteria for thermal equilibrium. Unless transition

probabilities can be formulated in this way, a relationship

betweenMonte Carlo time and real time cannot be clearly

demonstrated. n man y Monte Carlo studiesof time-depen-

dent phenomena, esults are reported in terms of integral

Monte Carlo steps,which obfuscatea definitive role of time.

Ambiguities surrounding the relationship of Monte Carlo

time to real time preclude igorous comparisonof simulated

results to theory and experiment, needlessly estricting the

technique. Within the past few years, the idea that Monte

Carlo methods can be utilized to simulate the Poissonpro-

cesshas been advanced n a few publications”* and some

Monte Carlo algorithms which are implicitly basedon this

assumptionhave beenutilized.lS4 his is an attractive pros-

pect, since within the theory of Poi ssonprocesses,he rela-

tionship between Monte Carlo time and real time can be

clearly established.

In this paper, we shall focus on dynamical interpreta-

tion of the Monte Carlo method . We shall show that if three

criteria are met, namely, that transition probabilities reflect

a “dynamical hierarchy” in addition to satisfying the de-

tailed-balance riterion, that time increments upon success-

ful events are formulated correctly in terms of the micro-

scopic kinetics of the system, and that the effective

independence f various events can be achieved, then the

Monte Carlo method may be utilized to simulate effectively

a Poissonprocess.Within the theory of Poisson processes,

both static and dynamic propertiesof Hamiltonian systems

may be consistently simulated with the benefit that an exact

correspondence etweenMonte Carlo time and real time can

be established n terms of the dynamics of individual species

comprising the ensemble.We shall demonstrate he formal-

ism by considering the approach to and the attainment of

Langmuir adsorption-desorption equilibrium in a lattice-

gassystem.We shall also discussstraightforward extension

of the methodology o more complicatedsystemsof interact-

ing particles.

II. DYNAMICAL INTERPRETATION OF THE MONTE

CARLO METHOD

Under a dynamical interpretation,’ the Monte Carlo

method provides a numerical solution to the Master equa-

tion

1090 J. Chem. Phys. 95 (2), 15 July 1991

0021-9606191 I 41OSO-07$03.00

0 1991 American institute of Physics

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mu,t)

- = 2 W(cr’-u)P(a’,t) c W(u-+o’)P(a,t),

dt -’

a’

(1)

whereu and u’ are successive tatesof the system,P( u,t) is

the probability that the system s in state u at time t, an d

W( u’ -+a) is the probability per unit time that the system

will undergoa transition from state u’ to state u. The solu-

tion of the Master equation s achieve d omputationally by

choosing andomly amon g various possible ransitions to a

model systemand acceptingparticula r transitions with

ap-

propriate probabilities.Upon eachsuccessfulransition (or,

in some nstances, achattempted ransition), time is typi-

cally incremented n integral units of Monte Carlo steps

which are related o some unit time r. At steady state, the

time derivative of Eq. ( 1) is zero and the sum of all transi-

tions nto a particular stateu equals he sumof all transitions

out of state u. In addition, the detailed-balance riterion

W(u’+o)P(u’,eq) = W(u-+u’)P(u,eq),

in which

(2)

P( a,eq ) = Z - ‘e - H(“)‘kBT,

(3)

must be imposed for each pair of exchanges, o that the

Monte Carlo transition probabilities can be constructed o

guarantee hat the systemwill attain a thermal equilibrium

consistentwith the model Hamiltonian. In Eq. (3), Zis the

partition function and N is the Hamiltonian of the system.

The detailed-balance riterion doesnot, however,uniquely

specify heseprobabilities. When static propertiesof model

Hamiltonian systemsare sought, Eqs. (2) and (3) are the

only standardswhich must bemet n addition to the necessi-

ty of a sampling procedurewhich is sufficiently random to

prevent statistical bias. Dynamical properties equire hat a

more definite relationship between he Monte Carlo time

step and the transition probabilities s established.We shall

show hat this relationshipcanbe established nd mple men-

ted, once ransition probabilitiesare ormulatedas ateswith

physical meaning, hrough the theory of Po issonprocesses.

In a dynamical nterpretation of the Monte Carlo meth-

od, t canbe assumedhat time resolution s accomplished n

a scaleat which no two eventsoccur simultaneously.Once

this perspectivehas been adopted, the task of the Mon te

Carlo algorithm is to createa chronologicalsequen ce f dis-

tinct eventsseparated y certain interevent imes. Since he

microscopic dynamics yielding the exact times of various

eventsare not mo deled n this approach, he chain of events

and co rrespondin g nterevent times must be constructed

from probability distributions weighting appropriately all

possibleoutcomes.The distributions governing ransitions

and nterevent imesavailable o a systemat any ime t canbe

develop ed rom considerations ully consistentwith the me-

soscopic genre. On a course-grained,mesoscopic evel, it

must be assumed hat the totality of microscopic nfluences

underlying various transitions of a system dictate certain

distinctive eventsE={e,,e2,...,en 1, which can be character-

ized by average ransition ratesRsir,,r,,...,r,, 1. In the ab-

sence f microscopicdetail, t can beheld hat any particular

transition which becomes ossibleat time c can potentially

occur at any later time t +

At

with a uniform probability

K. A. Fichthorn and W. H. Weinberg: Dynamical Monte Carlo simulations

1091

which is basedon its rate and is independentof the events

before time t. Let us consider the ramifications of these

premises or a stationary processwith two states epresent-

ing “forward” and “reverse” transitions. This processmay

correspond,e.g., to the time-depe ndentoccupancy or va-

cancy of one “site” amon gmany in the adsorption-desorp -

tion equilibrium of a gas-pha se oleculewith a solid, single-

crystalline surface. Adopting a frequency definition of

probability, the average ate t of, say, he forward transition

(which become s ntermittently available via the reverse

transition) can be interpreted as a time density of events.

Sampling small, identical time intervals S of a larger time

increment, =

na, the average ate s the ratio of the number

of time intervals containingeventsns to the total number of

intervals sampledn per unit time S in the limit S-0 and

n-co,

r= lim 2.

(4)

&.o,r-cc t

In the limit S-O, each nterval will contain, at most, one

event. Also, consistent with our premise s that each ime

interval has an equal probability r8 of containing an event.

Let Ne,tbe a randomvariablecounting he numberof events

which have occurred within a time t. Then, the probability

that n, eventswill occur in time t is

PWe,t

=n,) =

0

f- (rS)“‘(l -rS)“-“‘,

(5)

c

and n the limit S+ 0,

f’(N,, =n,) =(rt)“‘e-“.

n,

From Fq. (6), the expectednumbe r of events occurring

within a time t is (N,,,) = rt, from which the rate is recov-

ered hrough division by t. In Eqs. (4)-(61, it is seen hat a

mathematicaladaptionof these ery basicassumptionseads

to the characterizationof a stationary series f random, nde-

pendenteventsoccurring with an average ater in terms of a

Poisson rocess.

*10t can be shown hat a Poisson rocesss

consistentwith the Master equation.” Additional features

are attributable to the Poissonprocessand the most signifi-

cant of these or the purposeat han d is characterizationof

the probability density of times t, between successive

events”

f,,(t) = re-“. (7)

From the probability density, he mean ime periodbetween

successive vents s calculatedas (t, ) = l/r.

A particularly useful feature of the Poi ssonprocess s

that an ensemble f indep endentPoissonprocesses ill be-

haveasone, argePoissonprocess uch hat statistical prop-

erties of the ensemble an be formulated n terms of the dy-

namicsof individual processes. onsider ingN-independent

forward-reverse Poisson processes or, keeping with the

previousanalogy, he adsorption-desorptionequilibrium of

an entire systemof independ entmolecules),eachwith some

arbitrary, but finite rate ri, let N,,, be a rand om variable

counting he overall numbe rof events n the ensembl e hich

haveoccurredwithin a time interval t. The quantity N,,, is

then the sum of rand om variablescounting the numbe r of

J. Chem. Phys., Vol. 95, No. 2,15 July 1991Downloaded 07 May 2001 to 128.174.129.95. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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events which have occurred in each of the individual pro-

cesses,.e.,

Neat = 5 New

(8)

i=l

The overall probability of n, events n time

t

is given by the

convolution of the individual probability mass functions

characterizing each of the individual processes

Here, 19s the fractional surface coverageof A. The absence

of adsorbate-adsorbate nteractions in our example allows

Eq. ( 12) to be solved readily for the approach to and the

attainment of adsorption-desorption equilibrium once r,

and rD are known in terms of the intensive and extensive

properties of the system.

With the initial condition

@t=O) =o,

e(t) = yr (l-e-“~+rq

(13)

A

D

RN,,, = n,1 = P(N,,,,)*P(N,,,,)*...*P(N,,,) , (9)

and the probability mass unction characterizing the overall

distribution of events s obtainableby the method of charac-

and, in the limit as + ~0,

teristic functions’*

rA + rDr

P(N,,, =n,) =@&

--It

n,

,

where

A= 2 ri.

i= I

(11)

A final point which should be emphasized s that the

basicpremises eading to Eqs. (6)) (7), and ( 10) are equally

applicable o systemswhich are nonstationary and evolving

toward equilibrium. In the nonstationary Poisson process,

the overall rate simply becomesa function of time. Thus, if

the Monte Carlo algorithm can be made o simulate he Pois-

son process, hen the relationship betweenMonte Carlo time

and real time can be given a firm basis n both static and

dynamic situations. In the following section, we shall dem-

onstrate, through an example of the approach to the attain-

ment of Langmuirian adsorption equilibrium, the criteria

which must be applied so that the Poisson processmay be

effectively simulated with Monte Carlo methods.

III. ADSORPTION-DESORPTION EQUILIBRIUM

Adsorption equilibrium of a gas-phase pecies with a

solid, single-crystalline surface occurs when the chemical

potentialsof gas-phase nd chemisorbedA are equal. From a

kinetic point of view, adsorption equilibr ium (steady state)

is establishedwhen the net rate of chemisorption of gas-

phaseA is equal o the net rate of desorption of chemisorbed

A to the gasphase.Within the context of a lattice-gas model

in which adsorption is unactivated, each chemisorbed A

molecule requires one adsorption site. We assume hat che-

misorbedA moleculesdo not interact appreciably with one

another and envision a collection of gas-phasemolecules

whose temperature, pressure, and intrinsic partition func-

tions dictate a seriesof independentarrivals of molecules o a

surface containing a uniform and periodic array of adsites.

The arrivals occur at random, uncorrelated imes and can be

characterized by an average rate r,. A similar scenario is

applicable o moleculeschemisorbedon the surface-the to-

tality of microscopic influences (e.g., surface phonons and

electron-hole pair creation) acting on an ensembleof che-

misorbed molecules nduces desorption events which occur

with an average ate r, . The appropriate kinetic expression

for this balance s

-$-=r,(l-0) -r,t9.

(12)

1092


(14)

Equation ( 14) reflects the detailed balance of the simple

adsorption-desorption model. If the rates are cast n terms of

appropriate partition functions,‘3-‘5 then both the desired

kinetics and thermal equilibrium of the system are ensured.

To maintain generality, we shall retain generic rate expres-

sions for the elementary steps. It should be stressed,how-

ever, that both kinetic and equilibrium behavior can and

should be ncorporated in the rates of elementary steps ndi-

genous o a particular system, so that both aspectsof the

system can be modeled consistently.

Figure 1 depicts the general features of one algorithm

for simulating as a Poissonprocess he adsorption equilibri-

um of a gas-phase peciesA with a two-dimensional lattice

containing N sites.A trial in this algorithm beginswhen one

of the N sites s selected andomly. If the site is vacant, ad-

sorption occurs with probability W, ; and desorption occurs

with probability W, if the site s occupied.Time is advanced

by an increment ri upon successful ealization of an event at

trial i and, for illustrative purposes,we shall also count the

overall number of trials

T,

which accumulate over repetition

of the algorithm. We shall show hat through a proper defini-

tion of W, and W, , the utilization of an appropriate ri , and

the random selectionprocess, he Monte Carlo algorithm of

Fig. 1 simulates he Poissonprocessand provides the correct

FIG. 1. A flow diagram for simulating as a Poisson process the approach to

and the attainment of Langmuirian adsorption equilibrium. 7’is the (inte-

gral) number of trials, f represents real time, ris a uniform random number

between 0 and 1, W, is the transition probability for event i [i

= A

(adsorp-

tion) or D (desorption) 1, and rr is the real time increment at trial T.

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solution to Eq. (12) for the N-site ensemble.

The first criterion which must be met to simulate effec-

tively the Poissonprocess s that the transition probabilities

W,, and W, must be chosenso hat the Monte Carlo simula-

tion obeysdetailed balance,asspecified,e.g.,by Eq. ( 14). To

demonstrate he manner n which this is achieved, et us first

consider the discrete stochastic processof the Monte Carlo

algorithm of Fig. 1. By performing this algorithm, we simu-

late a sequence f independent Bernoulli trials in which the

probability per trial of a successfuladsorption event is

WA

( 1 - 19~, the probability per trial of a successfu ldesorption

event s

W,

8, and the total probabil ity of success er trial is

WA ( 1 - 6Ji + W, 19~ 1. Here, 19~s the fractional cover-

ageof A at trial i, which for Mi occupied attice sites s given

by ei = M,/N. When the system has reached steady state,

(8, ) + t9, he simulated equilibr ium fractional surfacecover-

age of A. In general, 0, will be different from 0, the contin-

uum equilibrium fractional surface coverage,since he con-

tinuum coverageshould arise from time-weighted (and not

trial-weighted) Bi. However, in systems which are suffi-

ciently large, 8, should approach

8, (uide infra)

. The statis-

tics of the equilibrium system may be obtained readily. Let

NA,r be a random variable counting the number of success-

ful adsorption events n

T

trials. Then the averageprobabili-

ty of nA successfuladsorption events n T trials is given by

P(N,,,,=n,)=(T/n,)[W,(l--8,)]“”

X[I- wA(i-es)lT-+,

(15)

for O(n, <T. From Eq. ( 15), the expected number of ad-

sorption events n T trials is given by

(N~.~) = w,u -e,u-.

(16)

Similar results may be obtained for desorption events and

the overall statistics of both adsorption and desorption.

When steady state has occurred in the simulation, the

average ate of adsorption [obtained from Eq. ( 16) ] is equal

to the average ate of desorption, i.e.,

W,(I -es) = w,e,.

(17)

Detailed balance s satisfied at equilibrium (steady state) if

WA and W, are defined n a way which allows the physical

model representedby Eq. ( 14) to be recovered from Eq.

( 17). The transition probabilities satisfying this relationship

are not unique and, as noted previously, in the traditional

equilibrium application of the Monte Carlo method, these

probabilities are often formulated without regard for the be-

havior of the system away from equilibrium. To simulate

dynamical phenomena,an additional criterion is necessary

so that transition probabi lities reflect unique transition ra tes

(and, hence,simulate dynamics). Theseprobabilities should

be formulated so that a dynamical hierarchy of transition

rates is established n terms of appropriate models for the

rates of microscopic events comprising the overall process.

Generally stated, a dynamical hierarchy of transition proba-

bilities is created when these probabilities are defined, for a

transition i, as

wi = ri/&., 9

(18)

where ri is the rate at which event i occurs and gm,, &sup

{ti}. A dynamical hierarchy is not achieved,e.g., n the stan-

dard Metropolis algorithmI applied to systemsapproach-

ing equilibrium, because ll transitions of the system o low-

er or equivalent energy states are considered to have a

probability of unity. The Kawasaki transition probabili-

ties “-19 on the other hand, create a dynamical hierarchy

among transition rates. However, it has been pointed out3s4

that this hierarchy is not appropriate for most physical pro-

cesses.n the algorithm of Fig. 1, e.g., the transition proba-

bilities could be constructed through normalization of the

rates of adsorption and desorption by the larger of the two.

If, say, , > rD, then transition probabilities could be defined

WA=1 and W,=z,

rA

(19)

(i.e., iL,,

= r, ). With these definitions, the relative fre-

quenciesof adsorption and desorption events n the Monte

Carlo simulation will satisfy the detailed-balancecriterion

[ Eq. ( 14) 1 with 0, + 8, (uide infra). Furthermore, with this

choice of transition probabilities, the success-to-trial ratio

will be optimized for an algorithm such as the one depicted

in Fig. 1. It should be noted, however, hat whi le this partic-

ular algorithm is reasonably effective if the time scales of

various processesn the system are similar, its efficiency de-

clines as he stiffnessof the system ncreases. n stiff systems,

where the majority of eventsare generally confined to a mi-

nority of sites,many trials will have o be attempted before a

successfulevent s selected.Other, more efficient algori thms

are availableL~20~2’nd should be utilized to simulate these

systems.Regardlessof the implementation, the relative fre-

quencies with which various events are performed must

comply with the detailed-balance ondition for both the dy-

namics and the equilibrium of the physical system if any

meaningful comparison of the simulation of a physical sys-

tem is intended.

A second criterion which must be satisfied o simulate

the Poisson process is proper correspondence of Monte

Carlo time to real time. To this end, it should be noted that

the Poisson process s, in actuality, a continuous-time ver-

sion of the discrete Bernoulli processwhich is simulated by

the Monte Carlo algorithm when time is measured n terms

of trials, By replacing the discrete nterevent times with ap-

propriate continuous values, the Monte Carlo algorithm

produces a chain of events which is a Poisson process.The

continuous interevent times are constructed through the de-

velopments eading to Eqs. (7)-( 11). Upon each trial

i

at

which an adsorption or desorption event is realized, time

should be advanced with an increment ri selected rom an

exponential distribution with parameter

ri = (N- Mi)rA f Afir,.

(20)

Here, Mi is the number of sites occupied at trial i (i.e.,

ei = Mi /N) . The selection of a time increment in this way

yields consistencywith Eq. (7)) which provides he distribu-

tion of interevent times for the Poisson process.Over many

successful rials at steady state, the average ime between

successive vents s

(to)=q (N-&f,; +&f.r .

(21)

I A I D

Here, h is the fraction of successful rials at which Mi sites


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are occupied. Equation (2 1) representsa time weighting of

various configurations of the simulated N-site ensemble

which is consistent with that dictated by the detailed-bal-

ance criterion for the equilibrium ensemble.A finite-size at-

tice can approximate the continuum ensemble o the extent

that its sizeallows resolution of the ensemble. n general, he

simulated ensemble t a particular point in time [character-

ized, n our simple example,by t9, t) ] could fluctuate about

the true continuum ensembleat that time [8(t), in our ex-

ample without ever achieving exactly the continuum value.

Thus, time-weighted averages (which correspond to ther-

modynamic averagesat equilibrium if the detailed-balance

criterion for thermal equilbrium is fulfilled) must be com-

puted to estimate he true continuum ensembleat any point

in time to within the desired degreeof accuracy.

When time is incremented according to the procedure

delineatedabove and transition probabilities are chosen o

satisfy he detailed-balance riterion, the correct macroscop-

ic rates of adsorption and desorption can be measured rom

the simulation at steady state. Let us consider, e.g., the rate

of adsorption. Over Ssuccessful rials resulting in either ad-

sorption or desorption, (N,,s) adsorption events will have

occurred, on the average,where

W4s) = [ w (yy ;;Iy,B, ] s

A -s

=

[

T W”fl? r)J;lxei] s*

(22)

The correspondingamount of time which has passed uring

the S successes t is

A

i IA(N-Mj)D”i

.

Defming the rate of adsorption as he number of adsorption

eventsoccurring per site per unit time and utilizing the defin-

ition of IV,, and W, in Eq. ( 19) [or any definition satisfying

Eq. ( 18) 1, the simulated rate of adsorption (R, ) becomes

1094


a lattice sufficiently large to represent he full systemensem-

ble and to simulate independent events. Nevertheless, f

transition probabilities are chosen o satisfy the detailed-bal-

ancecriterion and f time is ncremented n a procedureanal-

ogous o that outlined in Eqs. (20) and (2 1) in thesesitua-

tions, the time series an be nterpreted in terms of a Poisson

processof one of the events (analogous to the continuum

rate-limiting step approximation), and accurate ensemble

averages an be obtained hrough time averaging.For exam-

ple, in the single-site adsorptiondesorption process ntro-

duced previously, successive dsorptions and desorptions

are correlated. Nevertheless,a Poisson processcan be con-

structed consis ting of one of the events,e.g., adsorption oc-

curring with a rate r, ( 1 - 8). The criterion of a random

selectionof lattice sites or potential eventspreventscorrela-

tions from developing among specific sites and is usually

achievedby utilization of an adequate andom number gen-

erator. In certain applications, the intersite correlations in-

duced by an inadequate andom number generator may lead

to erroneous esults.” The selection of an appropriate ran-

dom number generator s, therefore, an issue equiring care-

ful consideration.

Thus, through the example of the lattice gas, we have

outlined the basicelementscomprising a formalism through

which Monte Carlo simulations may be utilized to simulate

dynamical phenomenawithin the context of the lattice-gas

model. We have utilized this methodology to simulate the

adsorptiondesorption algorithm of Fig. 1. Simulations

were run on 128X 128 square lattices with r, = 1.0

(site s) - ’ and r, = 2.0 (site s) - I. Transition probabilities

were defined by normalization of each rate by the rate of

desorption (i.e., W, = l/2, IV, = 1.0). Figure 2 depicts

the fractional surfacecoverageof adsorbateas a function of

time for an initially empty surface or both the transient ana-

lytical (exact) solution of Eq. ( 13) and the Monte Carlo

simulation. The Monte Carlo curve is the result of one run

(K4) =p”u -eiv, =r,(l---8,),

i

where

(24)

Af;:/[r" (N- Mi) + r&f,]

4i = ~jif;./[r"(N-il f,) +rDMj]

Ari

=-

(to>

(25)

provides he time weighting of eachconfiguration character-

ized, in this simple system, by 19~. similar result can be

obtained or the desorption events.

/ 1.2 I , I

A final and perhapsmore subtle criterion which must be

fulfilled for the Poissonprocess o be simulated effectively s

that independence f eventscomprising the time sequence f

the process s achieved. Strictly speaking, the formal ism

which we have presented s valid only when independent

eventsare simulated. n general, he ndependence f succes-

sive rials is ensuredboth by utilization of a systemwhich is

sufficiently arge hat single site and ntersite correlations are

lost, and by random selection of sites on which potential

eventsmay occur. The former condition is not always possi-

ble to achieve,particularly if the system s stiff. In such sys-

tems, t may not always be feasible omputationally to define

0.0 0.5 1 .o 1:5 2.0 2.5

t s

FIG. 2. The transient solution of Eq. (12) for the initial condition of an

empty surface [ 0( t = 0) = 0] provided by both the analytical form of Eq.

(13) and the MonteCarlo algorithm of Fig. 1 with r, = 1.0 (sites)-’ and

r,, = 2.0 (site s) - ’ ( WA = l/2 and W, = 1). The inset depicts the rate of

adsorption measured from the Monte Carlo simulati on at steady-state and

the continuum steady-state rate.

J. Chem. Phys., Vol. 95, No. 2,15 July 1991


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only. There is excellent agreementbetween he two solutions

in the approach to equi librium and the steady-state frac-

tional surface coverage.The inset of Fig. 2 depicts both the

analytical (exact) and Monte Carlo steady-state ate of ad-

sorption (desorption) . The rate of adsorption on a per site

basiswas measuredas he reciprocal of the variable amount

of time At required for 50 adsorption events (i.e., r, = 50/

NAt,

and N is the size of the lattice). It can be seen hat this

rate fluctuates about the predicted continuum rate of

r,

= 0.666... site s) - ‘. Of course, he amplitude of the fluctu-

ation dependson the number of eventsaveraged.The mean

rate measured rom the simulation over the depicted time

interval of 2.0 s was 0.67 f 0.06 adsorptions/site/s.

IV. GENERAL APPLICABILITY OF THE FORMALISM

Monte Carlo simulations are preferable o macroscopic

continuum approaches whenever structural or configura-

tional properties

of

a system arise which cannot be approxi-

mated analytically with a reasonabledegree of accuracy.

Such spatial organization occurs in a surprising number of

simple systems2*23-27n which moleculesmay be considered

effectively to be hard spheresand potential energy surfaces

are otherwise uniform. In systemssuch as hese, he identifi-

cation of characteristic time scalesand their incorporation

into a dynamical Monte Carlo algorithm is accomplished

through a straightforward extension of the methodology.

Hence, direct and unambiguous comparison of simulation

results o theory is possible.Perhapsdynamical Monte Carlo

simulations will be of the greatest utility when they are

linked in a hierarchy with

ab

inifio quantum mechanicsand

molecular dynamics to model the dynamics of systems of

molecules which interact with one another. When concen-

tration-dependent potential energysurfaceshavebeendelin-

eated,eventscan be definedas a molecule’s crossing he sad-

dle point between two potential energy minima and the

subsequent edefinition of the potential energy surface (due

to the local shift in concentration) which exis ted prior to

barrier crossing. Average rate coefficients or barrier cross-

ing of molecules n all possibleconfigurations of the system

can be calculated (for a lattice gas with localized interac-

tions extending only a few molecular diameters, he number

of distinct transitions will be imited) and incorporated into

a dynamical Monte Carlo simulation aimed at discernment

of global characteristics of the full system.Such calculations

involving semiempirical assumptions egarding the depend-

ence of the rate of barrier crossing on local concentration

through localized interactions have beenof utility in the in-

sight they have provided to thermal desorption,28-32he dy-

namics of surface eactions,28*33-39nd diffusion. 14.6*7*28*40*4’

The methodology may now be generalized.Let us con-

sider a systemcomprised of N specieswhich, considering all

possiblespatial arrangementsof the systemand correspond-

ing changes n the distribution oftransition rates, are capable

of undergoingk transition events.The k transition eventsare

characterizedby rates R = {r,,r, ,..., k} which are formulat-

ed to be consistentwith averagedynamics of barrier crossing

on some orm of potential energy surface.The N species an

be partitioned among the various possible ransition events

asN={n n

, 2,...,nk1, where ni is the number of species apa-

ble of undergoing a transition with a rate ri and

N= + n;.

i%l

(26)

Thus, a particular configuration of the systemat a particular

time can be characterized by the distribution of N over R.

This distribution is constructed by a Monte Carlo algorithm

which selects andomly among various possibleeventsavail-

able at each time step and which effects he events with ap-

propriate transition probabilities W = {w1,w2,...,wk . The

transition probabilities should be constructed in terms of R

so that detailed balance s achieved at thermal equilibrium

and a “dynamical hierarchy,” as expressed n Eq. ( 18)) of

transition rates s preservedaway from equi librium. If a suf-

ficiently large system s utilized to assure hat the indepen-

dence of various events s achieved, then the Monte Carlo

algorithm effectively simulates he Poissonprocess,and the

passage f real time can be maintained in terms of R and N.

To accomplish this, at each rial

i

at which an event is real-

ized, time should be updated with an increment 7i selected

from an exponential distribution, i.e.,

1

fi = -

----In (W,

Zi niri

(27)

where U is a uniform random number between0 and 1. This

procedure should ensure hat a direct and unambiguous e-

lationship betweenMonte Carlo time and real time is estab-

lished.

V. CONCLUSIONS

In conclusion, we have outlined the basic elementscom-

prising a self-consistent dynamical interpretation of the

Monte Carlo method. We have shown that if three criteria

are met, namely, if ( 1) a dynamical hierarchy of transition

probabilities is created which also satisfies he detailed-bal-

ance criterion; (2) time increments upon successfulevents

are calculated appropriately; and (3) the effective ndepen-

dence of various events comprising the system can be

achieved, then Monte Carlo methods simulate a Poisson

process, and both static and dynamic properties of model

Hamiltonian systemsmay be obtained and interpreted con-

sistently. We have demonstrated the methodology through

the example of the approach to and the attainment of Lang-

muirian adsorption equilibrium and discussedextension of

the methodology to systemsof interacting species.We anti-

cipate that dynamical Monte Carlo simulations will contin-

ue to be a valuable ool in the developmentof statistical me-

chanical theory and in modeling applications. The

formalism which we have proposed should, henceforth,

eliminate any ambiguity surrounding the conduction and

the interpretation of dynamical Monte Carlo simulations.

ACKNOWLEDGMENTS

This research was supported by the National Science

Foundation under Grant No. CHE-9003553. One of us

(K.A.F.) thanks IBM for a postdoctoral fellowship.


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