some generalizations of the trapping problem

PHYSICA ELSEVIER Physica A 215 (1995) 40-50

Some generalizations of the trapping problem A.M. Berezhkovskii a,1, George H. Weiss b,.

a Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, USA

b Physical Sciences Laboratory, Division of Computer Research and Technology, National Institutes of Health, Bethesda, MD 20892, USA

Received 3 December 1994

Abstract

We consider two effects on kinetics that have not generally been taken into account in analyses of classical trapping models for the reactions A + T ~ T and A + T --+ 0: the effects due to correlations in the initial positions of the A's and those due to the state of mobility of each of the species. Our analysis is formulated in terms of three-dimensional Brownian motion. We give a heuristic treatment of the short-time regime based on statistical properties of the Wiener sausage. The effects of initially correlated positions are modelled in terms of a set of a multiplicity of A particles located at the origin.

1. Introduction

The problem of determining properties of the number of distinct sites visited by an

n-step lattice random walk has appeared in many guises in the literature of both mathe-

matics, [ 1-3] and physics, [ 4 - 7 ] . Several more references to work in this field are to be

found in the review in Ref. [ 8 ]. One source of interest in this class of problems is that of

generalizing the s imple Smoluchowski model for the reaction rate in diffusion-controlled

reactions, [9,10]. An important feature of the original Smoluchowski formulation is its

neglect of many-body effects. The problem of accounting for at least one source of

many-body effects can be studied in terms of the so-called trapping model, [8] . In

the original version of this model a single random walker on a lattice is allowed to

* Corresponding author. 1 Permanent address: Karpov Institute of Physical Chemistry, UI. Vorontsovo Pole 10, 103064, Moscow K-64,

Russia.

0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 037 8-437 1 (95) 00009-7

A.M. Berezhkovskii, G.H. Weiss/Physica A 215 (1995) 40-50 41

wander through a field of randomly placed immobile traps, reaction being identified as

an encounter between the random walker and a trap. This is an idealized model for a

reaction which we represent schematically as A(mobile) + T(static) ---. T(static), the notation indicating which of the species is able to move. The trapping model therefore

replaces the single trapping molecule in the Smoluchowski model by an infinite number of traps.

A generalization of this class of models replaces the single random walker by N >> 1

random walkers. A development of the Smoluchowski theory in its original form requires

consideration of only a single random walker since Smoluchowski's original assumption is that there is no competition between A particles for traps. Since the survival probabilily

of any single random walker at step n depends on its avoidance of traps until that time

the problem of finding a functional form for the survival probability is equivalent to

determining the probability that each site visited by the random walker is not a trap.

Since some sites may be visited more than once, the kinetic behavior of the trapping

model will be determined by the number of distinct sites visited by the random walker. A

complete analysis of this random variable poses an extremely challenging mathematical

problem, [1,8], but it is relatively straightforward to determine an asymptotic form for

the expected number of sites visited by an n-step random walk. A recent paper contains an analysis of the expected number of distinct sites visited

in n steps by N random walkers that start from the same site on a lattice, [ 11]. An understanding of the kinetics of this class of models also sheds light on the effects

of initial correlations in position of the A's on the kinetics of the reaction scheme

A(mobile) + T(static) --~ T(static), at least within the framework of the Rosenstock

approximation, [ 12]. Let c be the probability that any given site is a trap, let R, be the

number of distinct sites visited by an n-step random walk, and let (Rn) be the mean of

this quantity where the average is taken over all walks. The Rosenstock approximation

can be stated in terms of these parameters as

S(n) = ((1 - c)R':> .~ (1 - c) <R') , (1)

where S(n) is the survival probability of a single random walker. Blumen, Klafter and

Zumofen have shown, by simulation, that this provides a lowest order approximation to the behavior of S(n) both on uniform and fractal lattices at small, but not uninteresting,

values of n, [ 13,14]. Higher approximations to S(n) derived in this spirit are available, but these require a knowledge of higher moments of Rn. Even a determination of the asymptotic form for (R. z) requires the use of very demanding mathematical techniques, [l] .

In the present paper we consider a number of generalizations of the Smoluchowski

model for the kinetics of diffusion-controlled reactions in three dimensions. A principal focus of our analysis is on the effects initially correlated positions of the A species.

Only the simplest model of such correlations will be considered, which locates the N particles initially at the same site. A second factor explored in the present paper is that of relative mobility, i.e. the differences that depend on which of the species is stationary

and which is mobile.

42 A.M. Berezhkovskii, G.H. Weiss/Physica A 215 (1995) 40-50

The classical version of the trapping model is formulated in terms of a single mobile random walker which is allowed to wander among randomly placed trapping sites. When the traps are mobile and a single particle, the equivalent of the random walker, is

stationary then the Smoluchowski solution for the survival probability is exact, [ 15,16].

There have been some recent studies in which both the traps and the single random

walker are mobile, [ 17,18], but far more exploration of this type of problem is required to understand characteristics of such models even to the limited extent to which those of

the standard model are known. In the present paper we discuss some partial results on

the kinetic behavior at short times of the following three-dimensional diffusion processes

with N >> 1 A particles and an initially uniform concentration of traps or T particles:

(a) A(mobile) + T(static) ~ T(static),

(b) A(mobile) + T(static) ~ 0,

(c) A(static) + T(mobile) ~ 0.

These are to be contrasted to earlier studies which concentrate mainly on properties in

asymptotic regimes in which scaling properties can be found rigorously, [7,8,19].

In case (a) the reaction kinetics are independent of the initial distribution of the A's. However, when the A particles are initially bunched, the rates of disappearance of A's

in models (b) and (c) will necessarily be less than in (a) because of the disappearance

of traps in the neighborhood of the initial positions during the course of the reaction.

The model in (b) is especially interesting because when the initial positions of the A's

are initially uncorrelated the Smoluchowski theory predicts that the disappearance rate

is independent of whether the A's are mobile and the T's fixed in place or vice versa. As mentioned, the Smoluchowski theory is exact when traps are mobile and the A's are

static. An interchange in the state of mobility gives rise to a difference in kinetics due to many-body effects [ 15,16]. It is these effects that are responsible for the decrease

in the particle annihilation rate as compared to that predicted by Smoluchowski theory. It will be shown in Section 4 that when the initial positions of the traps are initially

correlated the particle annihilation rate is lower when the A's are static and the traps are mobile.

The next three sections discuss in succession each of the cases that we have enumer-

ated and the final section compares some of these results. Our mathematical development

is based on determining stochastic properties the volumes of the Wiener sausages generated by Brownian particles, [21,22], as in [23], rather than on the lattice random walk. All of the derivations to be given relate to particles moving by Brownian motion in a continuum but simulations are made on a system consisting of a lattice random walk. We will assume that any A that approaches to within a distance a of a trap is instantaneously removed from the system. Whenever a trap is unchanged after an encounter one expects the kinetics of particle annihilation to be independent of how the A's are initially distributed. In the present paper we ignore many-body effects due to the competition of the traps for an A and estimate the kinetic behavior in the framework of the Smoluchowski or mean-field approach.

A.M. Berezhkovskii, G.H. Weiss/Physica A 215 (1995) 40-50

2. A(mobile) + T(static) ~ T(statie)

43

This case reduces to the standard trapping problem when N = 1. Let vl (t) be the

average volume of the Wiener sausage generated by a single Brownian particle of radius a, [23,24]. When the traps are initially uniformly distributed with a concentration c, the

survival probability of a single particle in the short-time limit can be written in terms

of Vl(t) as

S1 (t) = e x p [ - c v l (t) ] , (2)

[21 ]. The subscript 1 in the survival probability refers to the first of the three cases

indicated in the last section. In three dimensions the Smoluchowski analysis leads to an

exact formula for vl ( t ) , viz.

v l ( t ) = 4 7 r a D t [ l + 2 _ ~ ] , (3)

where D is the diffusion constant associated with the particle. The Smoluchowski result

in Eqs. (2) and (3) are known to accurate results in the simplest version of the

trapping problem at short, but still physically meaningful, times, [ 13,14], but not in the

asymptotic regime, [7,19 ]. The appearance of Eq. (2) can be simplified by expressing all

parameters in terms of dimensionless variables. We replace the time by the dimensionless equivalent ~- = D t / a 2 and the concentration of traps by the volume fraction ¢/, = 47ra3c/3, in which case Eq. (2) is written as

This is an expression for the survival probability of a single particle which, however,

does properly account for initial correlations when there is initially more than a single

A particle.

3. A(stat ic) + T(mobi le ) --~ 0

In this section we derive a formal expression for the survival probability of N immo-

bile Brownian particles in the presence of an initially uniform concentration of mobile traps. The A particles are assumed to be initially located at the origin. To simplify the analysis we will permit the traps to overlap. The rule used to define the dynamics of

the system is that when a single trap approaches to within a distance a to the cluster of A's, it disappears, at the same time removing exactly one A from the system. Let p(t lro) be the probability that a trap initially at r0 has not come into contact with the cluster of A's by time t. Therefore q(t[ro) - 1 - p( t lro) is the probability that it has

reached the cluster of A's by that time. Let us define a large volume which we denote by /2, which will initially contain an

average of M = f2c traps where /2 is so large that it can be assumed that M >> N.


Results will be derived in the limit ~2 ~ oo. Let a index the original configuration of traps and let the initial position of trap j in trap configuration a be denoted by rj(o~). The initial positions of the traps will be assumed to be uniformly distributed in /2. According to this assumption the probability that all M traps in configuration ce have avoided the point containing the A particles before time t is

M

p~(t) = IIP(t lr j(a) ) . (5) j=l

Since a is a single, randomly chosen, configuration the probability that none of the A's have been trapped by time t by any single trap in the ensemble is equal to Po(t) = (p,~ ( t)) where the average is taken over all possible initial configurations, or equivalently, over all values of a. When the traps are initially uniformly distributed in space this function can be written explicitly as

P°(t)= lim 1 f / f i x a~0o -~ff "'" P(tlrj)dMrj 12 /2 j=l

" [ }" = lim p(tlr)dr = lim 1 - q(tlr)dr

/2---*0o /2---*00 /2

=exp [-c a f q(tlr)dr (6)

However, the integral f/2 q(tlr)dr is just the average volume of the domain visited by a single spherical Brownian particle in time t. This can be identified with the function vl (t) defined in the last section for traps modelled as spherical particles of radius a, [23,24]. Therefore the limiting form of the probability given in the last line of Eq. (6) is just Sl(t) given by Eq. (2).

The probability that exactly i out of the N particles have been removed from the system by time t is equal to the probability that i out of the original M traps have reached the A region; that is

[/1 --, ( M ) q(tlr,dt , (7) Pi( t) = 2iln q( tlr)dt 1 - -~ 12 12

where i ranges from 0 to N. We may appeal to the Poisson approximation to the binomial distribution, [25], to find that in the limit shown, Pi(t) is

[ COl ( t ).______~] e_CVt ( t P i ( t ) = - [ C V l ( t ) ] i S l ( t ) . ( 8 ) i! i!

The probability that n out of the original N A particles are untrapped at time t is Qn(t) = PN-n(t) and the probability that none of the A's survive at time t is therefore given by


0 I I

0.40 1.10 1.80

© N =200

[ ] N ' 4 0 0

log(n)

Fig. 1. Two plots of ( I / n ) I n [ 1 - S(n)]2/3 as a function of In(n), together with approximating fits to a straight line for a trap concentration equal to 0.1 and for N = 200 and 400 particles. The validity of Eq. (17) is equivalent to exact linearity in the plot.

O(3

Oo(t) = Z Pi(t). (9) i=N

Hence the fraction of surviving A's is equal to

(N(t)) X-~ [ 1 _ / ] [CVl(t)] i S ( t ) = _

- - -/--' L N] i-----i~--- " e -CVl ( t ) . ( 1 O) N i=0

When N >> cvl (t), which means that the mobile traps have not had enough time to visit the origin which contains the A particles, this fraction is given by

S(t) ,,~1 cv,(t)__N ~ e x p ( CVl(t))N ' (11)

which is derivable from Eq. (10). In the opposite limit when the traps have been allowed to explore an extensive part of the space, i.e. N << cvl(t), the major contribution to S(t) is made by the term in the sum in Eq. (10) indexed by i = N - 1. This leads us

to the approximation

S(t) ~ [CVj(t)]N-le-CV~(t). (12) N~

We show, in Fig. l, the result for ln[ 1/S(n) ] obtained from a simulation based on 1000 A particles set in a field of traps with c = 0.9. Since vl(t) is approximately linear in the time one expects the curve to also be linear in the time variable. The data are seen to be in close agreement with this prediction.


4. A(mobile) + T(statie) ~ 0

In this section we calculate, by a heuristic mean-field argument, the kinetics of

decay of the N original A particles when traps and particles annihilate one another on

encounter. We will assume the traps to be distributed with a density c throughout space. As before we let ( N ( t ) ) denote the average number of A particles that survive till time t and let v( t ) be the cumulative volume visited by both the surviving and annihilated A particles up to that time. In three or more dimensions, on average, the rate of decrease of the A's will be proportional to the rate at which the cumulative volume is changing, which is equivalent to the relation

= d r ( t ) dIN(t)------~) - c (13) dt dt

When the initial positions of the A's are distributed sufficiently far away from one another in space, their positions being uncorrelated, each A particle will make an independent contribution to the kinetic behavior of the system in three or more dimensions. Hence an expression for IN( t ) / can essentially be found in terms of the survival probability of a single A. This, in turn, implies that the Smoluchowski theory provides a satisfactory initial approximation to the rate of disappearance of A's:

S( t ) ~ exp [ - c v l ( t ) ] , (14)

which is the solution that follows from a Smoluchowski-like argument. When, however, the initial positions of the A's are highly correlated this simple argument cannot be correct because of the significant overlap of Wiener sausages generated by different Brownian particles.

The time-dependent behavior of the average volume swept out by N spherical particles of radius a initially located at the same position is discussed in [26] where approximate results for this random variable are derived. This volume will be denoted by VN(t).

While this is not exactly the function that appears on the right-hand side of Eq. (13) it nevertheless can be used to provide an approximation to v(t) at sufficiently long times. In the following analysis we will use the dimensionless time r defined earlier. At long times, defined by the requirement r >> N 2, the initially correlated Brownian particles, to a good approximation, separate essentially completely because of the transience of three-dimensional Brownian motion. This allows us to write

dVN(r) ~ N dr1 7" >> N 2 . (15) dr dr '

In contrast, when r << N 2 the effect of the initial overlap persists and VN(~)(r) is only weakly dependent on N( r ) , i.e. it has the dependence

d~'N ( <N(T)> 2 ) N2 dr ~ - 3 v ~ r l / 2 ln3/2 \ 47rr ' 1 << r << , (16)

in which we have set vN(r ) = VN(r) /VN(O), where VN(O) = 4rra3/3.


At suffÉciently long times (N(7")) is approximated by the result provided by the

Smoluchowski theory, and given in Eq. (14). When traps are sufficiently rare so that

<< 1, particle annihilation takes place on a time scale ~- >> 1/¢. Hence at short times

we can set (N(T)) ~ N in Eq. (16) which allows us, as a first approximation, to write

[ S(~') - (N( 'r)) _ 1 CUN(r)(T) ~ 1 X/8¢ r ln U U - - - i f - '

1 < < T < < N 2 . ( 1 7 )

Since we have assumed that N is large, the time span over which this equation is

presumed to hold is also large. Hence the anomalous form in Eq. (17) should be detectable in simulations. The form of this equation indicates that the disappearance

of A's in this range of T proceeds more slowly than indicated by the Smoluchowski

analysis since Eq. (17) is not exponential. The slowdown is due to the significant

overlap of Wiener sausages which at the earliest times clears out a number of traps in the neighborhood of the initial point. Thus, the number of active traps is smaller than in the absence of overlap. At a dimensionless time of the order of N 2 the trajectory

of the A's tend to separate, the Wiener sausages no longer overlap, and the particles

then disappear independently. This, then, manifests itself as an exponential decay of the

surviving A's. Thus, the decay in this second regime goes like

S( t ) ~ e x p [ - c { v l ( r ) - V l ( T O ) } ] , r > > g 2, (18)

in which TO is a crossover time to the asymptotic regime in which the particles can be

regarded as being uninfluenced by one another.

We have tested the approximation in Eq. (17) by plotting the function ( l /n) . [1 - S(n) ]2/3 (for discrete numbers of steps) as a function of ln(n). When Eq. (17) holcls

this function should be a straight line. One expects Eq. (17) to be valid when n is relatively small, or equivalently, when S(n) is sufficiently close to I. In Fig. 1 we show

such a plot for c = 0.1 and values of n small enough to ensure that S(n) >_ 0.7. The

linear fit appears to be quite convincing in the chosen range. When the value of n is

increased the deviation from the straight line fit becomes quite noticeable. It is interesting to compare this result with those of Havlin et al., [20]. These

investigators studied the survival of N lattice random walkers in d dimensions when the random walkers were set initially at the origin. All other sites on the lattice are occupied by traps, each of which is annihilated whenever one or more random walkers reaches

its site. In that study the authors find that the average fraction of surviving walkers goes asymptotically like

S( t ) = l -- f ( - ~ / d ) , (19)

in which the scaling function f ( u ) is found as the solution to

d f = kd f_2/d( 1 _ f ) (20) du


c co

1.00

0.80

0.60

0.40

0.20

0.00

(c)

(b

I I I

20 40 60 80 100

n

Fig. 2. A comparison of survival probabilities for the three types of reactions considered here. These are (a) A(mobile) + T(static) --~ T(static) (i.e. the standard trapping model), (b) A(mobile) + T(static) ~ 0, and (c) A(static) + T(mobile) --~ 0. The curves are based on 1000 runs which include 200 A particles and c=0.1.

where ka is a constant that depends only on the dimension d. A comparison of Eq. (17)

with Eq. (19) indicates that the three-dimensional results agree with Eq. (19) up to a

logarithmic factor, although the resulting f ( u ) is not a solution to Eq. (20). Whether

the coincidence in forms of the scaling behavior is more than simply a coincidence is a

question that merits further investigation.

5. Discussion

In earlier sections of this paper we have provided an estimate of the time-dependence

o f the decay in the average fraction of untrapped Brownian particles for the three

processes enumerated in the Introduction when initial positions are correlated. There

would be no correlation effects in a mean-field description of the reaction A(mobile) ÷

T(static) --* T(static). In the remaining two reactions the initial correlations lead to a significant slowdown in the trapping rate, as illustrated in Fig. 2. The slowdown of the

decay of the annihilation reaction A(mobile) + T(static) ~ 0 relative to A(mobile) + T(static) ~ T(static) is an effect clearly attributable to trap annihilation.

The curves in Fig. 2 illustrate the dependence on n of the ratio of decay of the two processes just mentioned. The qualitative features indicated reflect the fact that the initial

set of particles behave independently when vN(t) exceeds a critical volume, VN, which is an increasing function of N. When VN is small compared to the average inter-trap distance which is of the order of c -t/3, effects due to initial correlations would be

minimized.


A point of special interest is the slowdown in the disappearance of A's in the reaction

A(static) + T(mobile) ~ 0 as compared to A(mobile) + T(static) ~ 0. This is a

measure of the slow rate required for initially randomly distributed traps to visit a small

region of space in three or more dimensions, as well as the fact that the underlying

diffusion process is transient rather than recurrent. This effect would presumably be

diminished in one, two or noninteger dimensions in which random walks are recurrent rather than transient. However, this point has not been investigated as yet.

There are many questions that remain open for further study in this general circle of

problems. In the present paper we have considered only the effects of initially correlated

positions of the A's in three dimensions together with some effects of which particles

are mobile and which are fixed in place. An intriguing problem along these lines is that of determining the kinetics of systems in which both particles and traps are allowed

to move. This proves to be difficult one to solve even when there is only a single

A in the system, [15,16,18], but such a system would provide a much more realistic

model for chemical reactions than those considered here. Further work is required to

understand the kinetic behavior when the traps are not initially randomly distributed but

are themselves correlated. It is known that when there is a single A in the system the survival probability can deviate from the classical Donsker-Varadhan form, [19]. We expect that corresponding deviations will occur when the system contains more than

one A particle. Unfortunately, to this point there are few analytical tools to address the

solution of such models, [27-33].

Acknowledgement

AMB is grateful to Dr. Attila Szabo for numerous very useful discussions related

to the problems considered in this paper and to the National Institute of Diabetes and

Digestive and Kidney Diseases for its hospitality during the course of this investigation.

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some generalizations of the trapping problem

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