Chemical Physics I55 ( I99 I ) 357-368 North-Holland

New Monte Carlo simulations of many-particle stochastic dynamics: growth of correlations and local self-ordering during annihilation of like particles

Ross Brown Centre de Physique Molhculaire Optique et Hertzienne, u.a. 283 du CNRS, VniversitP de Bordeaux I, F-33405 Talence Cede.xj France

and

Nikolai A. Efremov Institute of Spectroscopy of the Academy of Science ofthe USSR, 142092 Troitsk, Moscow Region, USSR

Received I February 199 I

This paper presents a new, general Monte Carlo method for simulation of the dynamics of a set of interacting and reacting particles, of several species, with individual events obeying Poisson statistics. The method is applied to the annihilation of like particles initially distributed at random on a cubic lattice and interacting by a tunneling law. The validity of some analytical results is related to the existence of several stages of the reaction with different small parameters. The simulations show growth of correlations and a tendency to local ordering of the surviving particles.

1. Introduction

The transport and reactions of electronic excitations in disordered solids continue to attract solid state phy- sicists and physical chemists, because of the importance, similarity and generality of the problems posed by such applications, as the conductivity and optical properties of disordered semi-conductors [ l-31, solid state chem- istry and photophysics [ 4-6 ] and relaxation in spin glasses [ 7-91. However, our understanding of even the simplest transfer processes is incomplete. This is related to the frequent absence of small parameters. The small parameters of such non-equilibrium systems must in fact change as the dynamics sweeps through different char- acteristic scales of time and distance, so that it is diff&ult to describe the whole evolution in the same way. In particular, growth of correlations between particles and in some cases a tendency to ~lf~rgani~tion [ lo- 121, produced by the dynamics, are possible. Hence, even the validity of known theoretical, approximate results may be difficult to decide.

Monte Carlo simulation of such problems is very useful for understanding them better and for improving the analytical results. Unfortunately, the rapidly growing volume of Monte Carlo work generally includes two kinds of simplification:

(a) The one-particle approximation, generally used to model transport processes, neglects interactions be- tween particles. Yet it is difficult to believe we can neglect correlations due to interactions or to competition for the same states, particularly since critical paths are a specific feature of disordered materials [ 13,141. These are (small) regions of higher than average density of the active sites, which must be used by many particles, even when the average carrier density is low. Such bottle-necks define microscopic or mesoscopic regions where in- teractions and hindrance can be important. This effect should be present even in iso-energetic systems. It must

0301-0104/91/$03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.

358 R. Brown, N.A. Efremov /Monte Carlo srmulations qfmany-parlicle stochastic dvnamics

be acute in inhomogeneously broadened systems at low temperatures [ 41. It is thus important to go beyond the one-particle approximation and to consider long-range forces, contrary to the second kind of approximation.

(b) Simulation of reacting or interacting particles commonly assumes nearest-neighbour interactions or even interactions only when particles are on the same site [ 15,161. This avoids difficulties stemming from the inter- action of background particles with any given pair, though it is clear that such interference must be important in real materials with long-range forces.

Simulation of reacting systems thus requires algorithms for the interaction of many particles with a given particle and for competition of many particles for a given final state.

In this paper we describe a new algorithm, which takes into account the above conditions, to directly repro- duce the stochastic, physical process by simulation. The algorithm can handle simultaneous processes of trans- port, trapping, annihilation (fusion), spontaneous decay and creation of several kinds of particles in disordered media or crystals. Excellent agreement was obtained with the known exact results on the Fiirster-Dexter transfer problem [ 171. New results will be published progressively. Some advantages and disadvantages of the new algorithm are compared with those of the “minimal process”, an old method of generating sample paths of a stochastic process [ 181. This earlier method was recently applied to a number of situations where the bookkeep- ing is formidable, such as birth and death and diffusion in one dimension [ 191, one-particle transport in inho- mogeneously broadened systems [ 41 and the diffusionless A+ B-0 reaction [ 201. The new algorithm is well suited to the large reactive systems which must be handled to exhibit the meso- and macroscopic ordering ex- pected above.

Here, we report on simulations of diffusionless annihilation, A+A+O, between randomly distributed A par-

ticles on a cubic lattice. Extension to continuum models, like glassy solutions, is simple. The results show a tendency to structural order. This behaviour illustrates a general feature of open, dissipative, non-equilibrium systems: the birth of structural order from an initially random state [ lo- 12 1. We also relate the validity of some analytical results to the existence of several stages in the dynamics, each with its own small parameter and scales of time and distance.

2. Static annihilation

We consider a set of A particles randomly distributed on a cubic lattice (lattice constant a= 1 ), with average

initial concentrationp, (0). Particles are removed by fusion, A+A+O, with an isotropic rate

w(r)=wo exp( -r/rO) (1)

for any two particles separated by distance r. The decay radius, r 0, depends on the overlap of the wavefunctions. Particles do not move. The dynamics obeys a hierarchy of equations for the 1, 2, 3, . . . . n, . . . particle distribution functions [ 2 1 1. The configurationally averaged probability that a particle survives at point r, at time t, i.e. the particle concentration, p, (r,, t), obeys the equation

dpl(rl, t) dt

=- Cw(lr,--r,I)p*(r,,r*,t), 12

where the two-particle function p2 (r,, r,, t ) is the probability of simultaneous presence of particles at r, and r2 at time t. It involves the three particle function and so on:

dp2(rl, r2, 0 dt

=--w(Ir,-r21)P2(r,,r2,t)- 1 (w(Irl-r31)+w(Ir2-r31))P3(rl,r2,r3,t)~ r,

R. Brown, N.A. Ejiiemov /Monte Carlo simulations of many-particle stochastic dynamics 359

As usual, the problem is finding a closed set of expressions for the average particle density, p, (t) and dp,/dt, which are measurable quantities. Agranovitch, Efremov and Zachidov [ 22 1, used a simple uncoupling approx- imation in eq. (3 ) (see also ref. [ 2 1 ] and references therein), setting

C (WI6 -431 +w(Ic--~3I )P3(ctr2,r3,Q 0

= c wj,r,_r3,)~2(rirr21t)~2frir~3,t) f Cw(fr2-r31)

kdrlt r2, fhtr2, r3,O

I3 PI (rlT 0 0 h(r2, t) .

The physical basis for this is low particle density and the sharp radial decay of expression ( 1) implying that particle 3 interacts with either 1 or 2, but not both. In fact, at time t, the reaction involves particles separated by distances close to the reaction radius [23] rJ t), such that w(r,)t QZ 1. See below for a more precise definition.

What may be the small parameters of this system? Let us first introduce

fX(t)=j &(t)p, (0) . (6)

a( f ) is in fact the mean number of particles which were initially in the reaction sphere. Approximation (5) should be good while the size of annihilating pairs is less than the average initial distance between particles, i.e. LU (t ) < 1. The first stage of the reaction should thus be a two-particle process, burning out the initial density ~uctuations, until the reaction volume reaches 1 /p, (0). But at this point, the particle density is lower, so rO( t) may still be less than the average distance between surviving particles. Let us therefore introduce a second parameter, the number of surviving particles in the reaction sphere,

B(l) =; ~~~(~)~~ (1) 1 (7)

where pi (t) is the average particle density. We thus may expect a second stage in the reaction. In this stage, the reaction should continue to burn out some new density fluctuations, in an essentially two-particle process. Dur- ingthisstage,cr(t)>l andjl(t)<l.

Burning out fluctuations implies that the reaction front should come closer to the average particle separation, F (t), where $tF3(t)pl (t ) = 1. This should signaf the start of the third stage of the dynamics, where many-panicle correlations should build up, probably leading to some structural order. If this is so, there should also be some spontaneous breaking of symmet~. Such behaviour is typical or open (the products of fusion are removed) non-equilibrium systems. Correlations in such systems are expected to grow both in value and in scale. We propose below a small parameter 6, which reaches 1 at the onset of the third stage. These ideas are illustrated below, both analytically and by simulation.

It is convenient to introduce the two- and many-panicle correlation functions defined by

p,(c, . . . . r,, t)=: fi PI Cri, t)&(rl t ... , r,, t) . (8b) ,=I

Introducing the g,, functions allows one to exclude effects of the particle density (included in pi(t)) and to concentrate on the arrangement of the particles. Thus, g,= 1 describes a homogeneous distribution, without correlations, in which ~uctuations are random. Deviation of g, from 1 can occur on finite scales of distance between the particles. g,c 1 describes the burning out of particles in the corresponding configuration, while g,> 1 describes a tendency to clustering beyond random fluctuations. In homogeneous, isotropic systems, the one-particle function is independent of position, rip and p2 and g2 depend only on the displacement, r= 1 r, - r2 1, Higher order functions, g,, would depend on n - 1 displacements, if the initial symmetry were conserved. This

360 R. Brown, N.A. Efremov /Monte Carlo simulations ofmany-particle stochastic dynamics

can be wrong in the third stage above, when many-particle correlations spontaneously break the symmetry. The description is then much more complicated. Alternatively, as below, we can use translationally and azimuthally averaged correlation functions. Note that in this case, even the isotropic, configurationally averaged g, may miss important, anisotropic effects of self-organisation, as in ferromagnetism for example.

Keeping this in mind, we can write eq. (2) differently as

dp, (t) -=- ~~(/r~-~~I)~~(r~~~)~~(r~,~)gz(r~,r~,f)

dt

= -nt)P:(t)

with a time dependent second-order rate constant

(9)

r(t)= C 4 Id k2( 14, t) . (10)

The particle density thus decays as

~,(t)=~,(O)l(l+p,(O) j- Ws)). 0

(11)

Further, inserting eq. (8) in eq. (3) and using eq. (2), one readily obtains the general relation below, where we keep the possible dependence of g, on r, and r,, to make the meaning clearer.

+ CP1(r3,t)w(lr, -r31)[g2(rl,r2,t)g2(rl,r3,t)-g3(rI,r2,r3,f)l I3

+ ~Pc(r3,f)w(lr2-sl)[8,( rl, r2, rk2(r2, r3, t) -g3(c, r2, r3, t) 1 . (12)

Approximation (5 ) is equivalent to neglecting the differences in the square brackets in eq. ( 12 ). Physically speaking, this means third particles r3 interact with the pair rl, r2, but are “seen” only by one or other of the particles at a time - three-particle and higher correlations are neglected. This is also the same as neglecting one of the correlations in the Kirkwood approximation [24], g3( Q, r2, r3, r)=gz(r,, r2)g2(rl, r3)g2(r2, r3) with gZ( r2, r3) = 1, for example, in the first sum in eq. ( 12). Eq. ( 12) then yields the “two-particle” or pair approximation:

g2(r,t)=ew[-w(r)fl . (13)

We stress that eq. ( 13) does not follow from simply putting g3 = 0 (or p3 = 0) in eqs. ( 3 ) and (4)) contrary to popular belief. It follows from complete neglect of third particles, or the special Kirkwood approximation above. Note that on the other hand, p2 (r, t) =pf (t)e- w(r)r always contains some contribution of third particles through

PI.

Then

r(t)= 1 w( Irl) e-“(i’l)‘, r

(14a)

or in the continuum approximation,

+CC

r(t)=4R s

dr r’w(r) e-w’(r)r. 0

(14b)

R. Brown, N.A. Efremov /Monte Carlo simulations of many-particle stochastic dynamics 361

Let us now examine the three stages of the reaction outlined above. In stage one, (Y (f ) < 1 and at very short times, w,t< 1, we have w(r)t< 1 andg z = 1 for all r and fusion occurs within a sphere of radius ro. Then

(15)

but this regime can be distinguishable only at high initial densities, P(0) c ro. This condition is difftcult to meet with the overlap of the wavefunctions in most real solids, so we examine below the practically significant low- density case, ~(0) > ro.

At long times, war =u 1, it is useful to distinguish in eq. ( 14) a “reaction front” function describing the distri- bution of reacting pairs, R( r, t),

Z?(r,f)= w(r) e-N’(r)r (16)

and to find the dominant length scale for fusion at any time. R(r) has the form of a narrow bell curve centred on the “reaction radius” ra( t). It is well represented by the following approximation [ 231 when r,( tf > ro:

R(r,t)=e-’ w(r,(t)) exp(-[r-r,(t)12/2a2), (17a)

l,a2= [w'(ra)12 w2trA

w”(rd +w,,tr jt --

w(th) ’ * (17b)

The tunneling interaction ( I ) yields r,(t) =ro In( wet), corresponding to

w(r,)t=: I, o=r,. (18)

Then the pair approximation ( 13) in eq. ( 14b) yields the following continuum approximations for Z”( t) and

P!(f),

ln*( w,t) r(t)=sxrlaw(r~(tf)r~(t)=4x~r:, t ,

PI(O) Pi (0) ‘let)= 1+~KIJr~In3(w,t)p,(0)X l++tr~(t)p,(O) ’

(1%)

(19b)

where q=,/%fex0.92. This approximation must be valid as long as cx( t) 5 1, when all initial pairs at r< F(O) will have burnt out. As

noted above, it should be reasonable in stage IX of the reaction as long as there is less than one particle in the reaction volume, p( t ) < 1. How long does stage II last? We might use eq. ( 19b) to estimate the time when the reaction front will catch up on the average separation, marking the expected transition to a new regime with correlations and some ordering, stage III. From approximation ( 17), it follows that

and we might conclude from this that /3( t) -+ 1. But this is physically strange, because if rJ t) = F( t), all particles annihilate simultaneously, which is impossible. Hence, we expect r, ( t ) /F( t) < 1, which implies that the denom- inator in eq. ( 19b) is too small and we conclude that the true g2 function must be greater than e-wfrfr at the reaction front. This situation can occur only if higher correlations are present, cf. the integrals in eq. ( 12). The sum of the integrals in eq. ( 12) is of the order of r( t)p, ( tfgz( r, t). The crossover to many-panicle correlations can be estimated as the point where the corrections dominate the first term on the reaction front, or

PI(t)~(t&(r,(t), t)==w(ra(f))i?2(ra(fh t), (21)

where P is the rate function corresponding to an estimate of the solution & of eq. ( 12):

362 R. Brown, N.A. Efremov /Monte Carlo simulations of many-particle stochastic dynamics

dk(r, 2) ---=--W(r)i?2(r~ t)+Pl(tV(t)t?2(r, t)

dt (22)

The solution of eq. (22) is

From eq. ( 9 )

mP,(o= - h-&,(t))=-: In p10 , 6 > l(O)

whence

and

Expression (23a) is an improvement on g2 since it is greater, but it is an overestimate of the true function, corresponding to the maximum error of neglected terms in eq. ( 12 ). In particular, g2 does not tend to 1 at long distances. See ref. [ 25 ] for a more sophisticated estimate of the correction in a similar problem. Condition (2 1) is equivalent to

d(t)=3 --$j (~w~(t)pI(0))=310-a(t)=1. r,(t)

This gives support to the idea that (I!(I) = 1 is too strong a condition and that the two-particle approximation may hold beyond (Y ( t ) = 1 (if ro/ra( t ) << 1). Note that at this level of approximation, we use the old estimate of

p, (t), because the radial decay of w(r) in the integral for r(t) reduces the sensitivity of p, (t) to errors in the correlation functions. For example, it can be shown that approximating g2 by a step from 0 to 1 at r= r, (t ) also gives the above result ( 17) for pi (t). This point is confirmed by the simulations below, where it is found that eq. ( 19b) is a fair qualitative approximation to p1 (1), despite gross errors in the pair approximation to g2, due to the appearance of higher correlations, i.e. stage three of the reaction. Analytical investigation of this stage and later ones is difficult. The simulations shed light on it.

Finally, the reaction rate, dpi ldt, is of interest, because it corresponds to measurable delayed fluorescence in disordered solid solutions of aromatic molecules. The A particles represent triplet excititations, with lifetimes longer than the present timescales. The A +A-+0 reaction corresponds to fusion in the triplet state leading to delayed fluorescence, in the case of a low yield of intersystem crossing from the singlet state back to the triplet state. Molecules with high yields of intersystem crossing would be better represented by the A+A-+A reaction, in which the product state is at random on either of the interacting sites. The same equations as above can be used for the second case, replacing w by 2w and p,, by pJ2”. This does not change qualitatively the preceding discussion.

In the present case, using eq. ( 19a),

ln’(w,t) s= -4xrzy P:(t) . (25)

At short times, p, (t) xp, (0) and the main dependence is expected to be close to a power law

R. Brown, N.A. Efremov /Monte Carlo simulations ofmany-particle stochastic dynamics 363

dp,% dt

ln2(wot) = t-'

t

2 Y<l,

while a ( t ) =K 1 and r, ( t ) > r,. At long times,

dp, 1

dt= t ln(w,t) xt-‘, y>l, (26b)

while a(t) z+ 1, the change in the value of y corresponding to the last stages of burning out initial close pairs. The validity of all these analytical results can be checked easily by comparison with simulation, to which we

now turn.

3. Numerical simulations

The stochastic dynamics of a system of particles can be described as a sequence of random events with random delays to reach the current state of the system. The next event and the distribution function of the delay generally depend on the state reached in the previous steps. But first suppose the system may be described by a sequence of independent events with waiting time distributions known in advance, e.g. a set of particles decaying inde- pendently with Poisson statistics. The waiting time distribution of the decay of the kth particle is then Vku,( r) = e-“lr where We is the decay rate. A possible sequence of events, or sample path of the system can then be found by the following steps:

(i) Generate a set of trial delays for all the possible events. (ii) Order the trial event times by increasing order. (iii) Do the events sequentially. Note that the 1st event is the one with shortest trial time, determined after step (ii), but before doing step

(iii). Clearly, an equally valid path would result if we did just the first event in step (iii) and then went back to step (i) for a new set of trial delays for the remaining events and so on. The second variant has, however, an advantage: it can be used for dependent events, such as in a system of interacting or reacting particles, if the distribution functions are updated after the first event, before going back to step (i). This leads to the following algorithm.

Algorithm:

( 1 ) Generate the initial state of the system. (2) Look at the current configuration of the particles at elapsed time t and draw a random number pk, uni-

formly distributed on [ 0,l ] , for each possible event Ek with associated rate wk. (3 ) Calculate the trial event times according to the individual Poisson laws, by setting TV= -1n (pk) / wk. (4) Look for the event with shortest trial time, Ek with g=min, (TV) and actually let it occur. (5 ) Update the configuration and the list of possible events and event rates and repeat the process from step

(2). The above algorithm directly mimics the stochastic process. The “minimal process” algorithm [ 18 1, applied

recently to some similar problems [ 4,15,19,20], has a different approach. The next even time is drawn from the exponential distribution of waiting times in the current state, w(T) =exp( - IkwJ), by setting T= -In(p) / &w~., where p is a random number on [ 0,l 1. Which event occurs is determined by the branching ratio. Interval [ 0,l ] is divided into segments of length & equal to the expected branching ratio of event E,, q&= wk/xjWp A second random number PE [ 0,l ] falls with probability & in the kth interval and shows which event is to occur next. Our algorithm has the advantage that the rate constant &w~ and the subdivision of [ 0, 1 ] into segments of length &, needed in the minimal process, have not to be updated at each step. On the other hand, the new algorithm requires many more logarithms. Despite this, the execution times were found to be comparable.

The minimal process and the new method give statistically equivalent results, as was checked in tests of over

364 R. Brown, N..-t. ~Tjkrno~ /.Wonre Carlo s:rnularions q/‘man_v-particle stochastlr dvnamics

70 million events, and as can be seen from the following arguments. From step (4) above, the next event time is

T=minl(rl,) .

But according to step (3 ), the trial delays TV are independent variables with exponential distributions:

P(Q>T)= exp( -wnT) . (27)

The probability assigned by the algorithm to the survival of the current state for a delay of at least T is thus

Y(T)=P(rl,>,Tforallk)= Qexp(-w,T)= exp(- F wkr), (28)

as in the minimal process. Further, from step (3 ), the probability that T,: lies between s and s+ds, for sa 0 and vanishing ds is

P(s~s~<s+ds)=w~exp(-wAs)ds,

while

(29)

(30)

Hence, P ( EA has the shortest trial delay, in interval [s,s+ ds] ), call it wk( s) ds is just

dsvk(S)=wLexp(-w,s) dsexp(- c w,s) If!-

(31)

The branching ratio of event E,, no matter what the event time, is thus

(32)

the same as in the minimal process. Note that a step or event in discrete simulation, like the minimal process or the current method, has nothing

to do with the fixed step in direct, step by step integration of the master equations of the stochastic process, e.g. the Chapman-Kolgomorov equations [ 18 1. Integration with a fixed step is inefficient, because of the stiffness introduced by the exponential coupling, eq. ( 1). Increments in the model time are irregular, because of the spread of the microscopic rate constants and of the probabilistic nature of the algorithm, directly mimicking the stochastic process.

4. Results

Data shown here correspond to about 1.5 x 1 O6 particles on 1000 samples of 26 x 26 x 26 sites. The calculation required about 30 h on a Sun 4165 workstation.

Fig. 1 a shows the decay of the average particle density, p, (t ), at an initial concentration of p, (0) = 0.1, with ~,/a= l/ln( 100) ~0.22, i.e. a low density case, since 7(0)/r,- -2.9. Statistical fluctuations were small, so

smoothing the decay, curve ( 1 ), was not necessary. The simulation shows waves due to the discrete lattice structure: The radial density of nodes, while rising on average as 4nr 2, has some holes. These are forbidden

separations for A particles, so the reaction momentarily runs out of fuel as the reaction front sweeps over them. Each hole thus produces a shoulder in the decay.

A second feature of the decay is its slowness: 1.5 decades of decay in 7 decades of time. There are two reasons for this. Firstly, the exponential relation ( 1) translates small increases in the particle separation, r, into large

R. Brown, N.A. Efremov /Monte Carlo simulations of many-particle stochastic dynamics 365

Fig. 1. (a) Decay of the average particle density, p,(t), on a cubic lattice, with p,(O)=O.l, r,=a/ln( 100). Curve (I ): Unsmoothed simulated decay; (2) and (3): Analytical results including two-particle correlations only, with lattice sum and continuum approxima- tion, respectively; (4) Neglect of correlations. (b) Ratio of the two-particle estimate of the density to the exact value, t(t), and the small parameters of section 2: cr( t), the number of initial particles in the reaction sphere;/?,(t) and,!I,( t), the true value and pair approximation to the number of surviving particles in the reaction sphere; S(r), the parameter (over)estimating the crossover to strong higher correlations.

increases in the characteristic time, 1 /w(r). Secondly, the depletion of population and the growth of inhomo- geneities, see below, slow down the kinetics, compared to classical chemical kinetics, in which diffusion helps to

maintain the local reactant concentration [ 171. Fig. 1 a also shows the two-particle approximations to p1 ( t ), curve ( 3 ) and (4 ), obtained by setting form ( 13 )

of gz in eqs. ( 11) and ( 14). It is very good at short times but visibly diverges (more than 5%) from the exact decay after times ~~1% 102. Fig. lb, shows a plot of E( f ), the ratio of the approximate result to the exact one, diverging suddenly from 1 to a sustained, slowly increasing value of about 1.4 in the time scale studied here. Note that the approximatep, (2) is larger than the exact value, in agreement with the discussion in section 2: the approximate r(t) (and g2) must be too small. r(t) was evaluated with a lattice sum, so curve (2) also shows shoulders, which are absent in the continuum approximation, curve (3).

The last curve of fig. la is the result of neglecting correlations in the hierarchy eqs. (3) and (4), i.e. g2= 1 (also the short time limit eq. ( 15 ) ) .

Let us now return to the small parameters of section 2. Fig. 1 b shows (Y, /3 and 6. The onset of error in the two- particle approximation is sharp and close to the point where a!(t) = 1, the end of stage I, in which initial fluctua- tions burn out. Fig. lb shows two values of /3. The first, /3,, is approximation (20) and the second, a, is the true value found by putting the true density in eq. (7). Whereas j$, rises to 1, which is unphysical (see above), p, levels off at z 0.5.

Although fi5 < 1, the approximation to p, (t) is constantly in error. Clearly the condition /3(t) < 1 is missing something as suggested in section 2. In fact, j?,< 1 because p, (t) decays faster than the expressions ( 19) in this time range. Fig. 2a, showing the reaction rate, dp, (f)/dt helps us to understand this. Since dp,/dr decays by seven decades while p: decays by three decades, we deduce from eqs. (9 ) and ( 10) that the main variation of the reaction rate is r(t), i.e. it is connected with the correlation function. Hence the error in p, comes from an error in the two-particle approximation, gz( r, t) = e-“‘(r)‘. Fig. 2b shows the exact value of the rate function r(t), its two-particle approximation and the over-correction P( t ).

The parameter B in fact stabilises (as does the error E (t ) ) after a (t ) = 1, close to the point where 6( t ) = 1. Between these points, stage II of the reaction, the error in the pair approximation grows from an initially small value. 6(t) = 1 marks the third stage, with fully developed effects of third and higher order correlations on g,, see fig. 3.

366 R. Brown. N.A. Efremov /Monre Carlo simulations of many-partrcle stochastic dynamics

Fig. 2. (a) Reaction rate, dp,/df, for the same parameters as in fig.

-x-tlti:;~:~:~~::

0 3

1. (b) ( 1): The exact r(t) deduced from the simulations; (2): Its two-particle approximation, eq. ( 19a); (3): The overestimate, eq. (23b).

Fig. 3 shows the developing error in g2 = e-“““‘. As the reaction front spreads, it burns a hole in the initially uniform pair distribution. In this, the first stage, the reaction removes close pairs and eq. ( 13) is a good approx- imation. Eventually, ra( t) reaches the initial particle separation, t(0) such that jrt~~(O)p, (0) = 1. Surviving particles then begin to have a choice of neighbours for reacting, (Y (t ) > 1, leading to the onset of many-particle correlations. Corrections to the pair approximation initially are small, stage II. During this stage a rim appears on the edge of the hole in gz, due to higher correlations. The rim grows during stage II, but stabilises at about the same time as /3 and E, when 6( t ) = 1, marking the beginning of stage III.

The rim means two things. The first is that the reaction front is enriched in particles, leading to a faster decay. The effect is only small because the exponential decay of w( r) in r(t) gives most weight to the inner edge of the rim.

Secondly, some kind of local ordering is setting in. This can be seen from the oscillations in g2 beyond the rim, which we insist are not statistical fluctuations. By analogy with liquids [ 261, the rim corresponds to the first shell of neighbours and the oscillations to the second and higher shells. In fact the rim and first and second oscillations correlate well with the peaks in the pair correlation function of randomly close packed spheres [ 271, in which hexagonal symmetry predominates. This result supports the idea of the breaking of symmetry (section 1 ), in our case a reduction from spherical to hexagonal symmetry. One kind of ordering, segregation in the A+ B+O reaction, is now well established [ 17 and references therein]. We believe the present report is the first example of breaking of symmetry and a tendency to self-ordering in a very simple (single species), well defined, open reactive system, studied from the microscopic viewpoint. The above ordering in three dimensions is dif- ficult to picture directly here, but is very clear in the results of fusion between randomly distributed particles in two dimensions. The general features of the above discussion also hold and will be presented separately [ 28 1.

We note in passing that the crossover to power law decay of the reaction rate, dp, ldt x tpr, r> 1, indeed cor- responds to the burning out of initial pairs, o(t) = 1, cf. section 2, when p, changes notably.

5. Conclusions

We have investigated the diffusionless A +A-+0 reaction by tunneling, for initially randomly distributed par- ticles in three dimensions, over an experimentally relevant dynamic range, p, (t)/p, (0) > l/40. A new Monte Carlo method and analytical methods were used and compared. The new algorithm is particularly convenient

R. Brown, N.A. Efiemos /Monte Carlo simulations of many-particle stochastic &zatnics 367

l1_‘/ w,t-IO3 I h

wet-IO5

0

/Jl_ , , , , , 3 6 90

Fig. 3. Pair correlation functions at increasing times, or reaction radii, cf. eq. (IS), shown by the arrows. Smooth curve, pair approxi- mation gr=e-“(‘)‘. Dots are the azimuthally averaged g2 values deduced from the simulations with error bars equal to the estimated standard deviation. Fine structure in gz. due to the discrete cubic lattice but irrelevant for our purposes is suppressed in the last plot, to show the oscillations more clearly, by averaging over an interval 6r=0.2a. First a hole in g, shows destruction of initial close pairs. Later a rim and oscillations appear due to the spontaneous lowering of symmetry during the reaction.

for many-particle reactive systems. It reproduces directly the stochastic physical process. Limitations of the analytical two-particle approximation to this problem were exhibited and explained. At least three stages of the reaction were shown to exist, with different small parameters. First, initial fluctuations bum out in an exactly two-particle process. Secondly, further, longer-ranged fluctuations burn out. Thirdly, the reaction front comes sufficiently close to the average separation of the survivors for each particle to interact potentially with several neighbours.

In the third stage, the simulations clearly showed qualitatively new behaviour for a simple reactive system: growth of correlations and spontaneous lowering of symmetry, both marks of self-ordering in open, non-equilib- rium systems. The pair correlation function showed peaks indicating local order like that in a liquid, with well defined first and second shells of neighbours around the reacting particles. A more detailed description of the next stages of the reaction requires a different mathematical formulation. The averaged two-particle function is not sufficient. This work is in progress.

368 R. Brown, N.A. Efremov /Monte Carlo simulations qfmany-particle stochastic dynamics

Acknowledgement

We thank Dr. M. Orrit for his stimulating interest in this work and Professors Ph. Kottis and R.I. Personov for their encouragement. NAE acknowledges the support of the CNRS.

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