PHYSICAL REVIEW E 72, 016127 共2005兲

First-exit times and residence times for discrete random walks on finite lattices S. Condamin, O. Bénichou, and M. Moreau Laboratoire de Physique Théorique de la Matiére Condensée (UMR 7600), Case Courier 121, Université Paris 6, 4 Place Jussieu, 75255 Paris Cedex, France 共Received 15 March 2005; published 25 July 2005兲 In this paper, we derive explicit formulas for the surface averaged first-exit time of a discrete random walk on a finite lattice. We consider a wide class of random walks and lattices, including random walks in a nontrivial potential landscape. We also compute quantities of interest for modeling surface reactions and other dynamic processes, such as the residence time in a subvolume, the joint residence time of several particles, and the number of hits on a reflecting surface. DOI: 10.1103/PhysRevE.72.016127

PACS number共s兲: 02.50.Ey, 05.40.Fb

I. INTRODUCTION

The theory of random walks on lattices is not only a beautiful mathematical object, it is also useful in numerous domains of physics 关1兴, including potential theory 关2兴, statistical field theory 关3兴, or biophysics 关4兴. Another natural application is the diffusion of adatoms and vacancies on a crystal surface 关5–7兴. Among the numerous issues involved in the study of such lattice random walks, one important area is concerned with random walks on finite lattices. There are two important reasons for that special interest. First, true physical systems are not infinite, so that explicit boundary conditions often have to be taken into account in order to properly describe situations in which confinement can be relevant. Second, exact solvable random walk problems in bounded domains are very rare, making this theoretical field an important problem in its own right 关8–15兴. Recently, Blanco and Fournier 关16兴 reported an important general result concerning the mean first-exit time of Pearson random walks 关17兴—that is continuous space and time random walks, with a given frequency of reorientation ␭ of the direction of the constant velocity ␷—in a bounded domain. They showed that the mean first-exit time of a random walk starting from the boundary of a finite domain is independant of the frequency ␭ of redirection, and is simply related to the ratio of the domain’s volume V over the surface S of the domain’s boundary. The corresponding equation is 共in three dimensions兲 具t典 = 4

V , ␷S

II. MODEL

Let us start with the definition of our model. First, we have a lattice, which may be of any dimension or connectivity: for instance, we can as well apply our results to the cubic three-dimensional 共3D兲 lattice than to the triangular 2D lattice. We study the motion of a random walker 共with memory: the random walker has a probability of switching direction ␭ each time it visits a site; note that we can go back to the model without memory by taking ␭ = 1兲. The random walker starts from the boundary of a domain 共see Fig. 1兲. The position of the random walker is denoted by rជ共t兲; the speed of the random walker is ␷ជ 共t兲. The only values it can take are the difference between the positions of two neighbors. The precise rules of the model thus write

共1兲

where ␷ is the speed of the walker. This result was extended by Mazzolo 关18兴 to the higher-order moments of the first-exit time, and by Bénichou et al. 关19兴 to general diffusion processes in a nonuniform energy landscape. In this paper, we show how these results can be extended to the important case of discrete space and time random walks on a finite lattice. In particular, we obtain very simple explicit expressions for mean exit times and mean residence times averaged over the surface of the considered domain, for rather general random walks. More precisely, the paper is organized as follows. In Sec. II, we define the model under study and the basic averages involved in the sequel. Section 1539-3755/2005/72共1兲/016127共9兲/$23.00

III is devoted to the study of first-exit time moments. In Sec. IV, we generalize this approach and explicitly calculate the mean residence time in a subvolume of the domain. Section V presents an analysis of the mean number of hits of a reflecting surface. In Sec. VI, we generalize the previous results to the very important case of a random walk in the presence of a potential. Finally, in Sec. VII, we consider the joint residence time of several particles in subdomains of the lattice and derive results that can be applied, for instance, to the theory of heterogeneous catalysis.

FIG. 1. A typical volume: The surface is the dashed line; to the right, the conventions for the surface.

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PHYSICAL REVIEW E 72, 016127 共2005兲

CONDAMIN, BÉNICHOU, AND MOREAU

␷ជ 共t + 1兲 =

再

rជ共t + 1兲 = rជ共t兲 + ␷ជ 共t兲 with probability 1 − ␭ ␷ជ 共t兲 random with probability ␭.

冎

共2兲 共3兲

We will also define conditional probabilities: p共rជ⬘ , ␷ជ ⬘ , t 兩 rជ , ␷ជ 兲 is the probability for the random walker to start from the position rជ, with the speed ␷ជ and arrive after a time t at the position rជ⬘ with the speed ␷ជ ⬘. We also note p共rជ , ␷ជ , t兲 the probability for reaching the boundary after a time t, when starting from the point rជ with the speed ␷ជ ; we have p共rជ, ␷ជ ,t兲 = 兺 p共⌺in,− ␷ជ ⌺,t兩rជ, ␷ជ 兲. ⌺

共4兲

The sum here means a sum over all the lines which cross the boundary, and to which we can associate a point ⌺in 共inside the boundary兲, a point ⌺out 共outside the boundary兲, and a speed ␷ជ ⌺ which points from the outside to the inside 共see Fig. 1兲. This convention means that we consider the random walker has reached the boundary as soon as it reaches the point inside the boundary, with a speed pointing out. This also means that the time needed to reach the boundary is equivalent to the number of 共not necessarily distinct兲 sites visited, not including the site the random walker begins on. Thus, with these probabilities, for any quantity ␸共t兲 depending on the first-exit time t we can define its average on t:

percubic lattice in dimension D, it is simply 2D. An example of such a volume average is 具t典V, which is the mean number of sites 共still not necessarily distinct兲 visited by the random walk 共starting from a random point in the volume兲 before exiting, not including the site it starts from. To finish this introduction, let us note a few important relations ¯␸共⌺in,− ␷ជ ⌺兲 = ␸共⌺in,− ␷ជ ⌺,0兲.

共10兲

This is because the random walker starting at this position automatically leaves the volume at time 0. We will often ⬁ ␸共t兲p共rជ − ␷ជ , ␷ជ , t + 1兲. We can have to compute terms like 兺t=0 notice that, if rជ 苸 V, then p共rជ − ␷ជ , ␷ជ , 0兲 = 0. Indeed, p共rជ , ␷ជ 兲 is equal to one if rជ = ⌺in and ␷ជ = −␷ជ ⌺, and zero else. From the definition of ⌺in and ␷ជ ⌺, we can see that there is no rជ 苸 V such that rជ − ␷ជ = ⌺in, and ␷ជ = −␷ជ ⌺. Thus, we have ⬁

⬁

t=0

t=0

兺 ␸共t兲p共rជ − ␷ជ, ␷ជ,t + 1兲 = 兺 ␸共t − 1兲p共rជ − ␷ជ, ␷ជ,t兲. 共11兲 Furthermore, we notice that

兺

rជ苸V,␷ជ

¯␸共rជ − ␷ជ , ␷ជ 兲 −

兺

rជ苸V,␷ជ

¯␸共rជ, ␷ជ 兲

= 兺 ¯␸共⌺out, ␷ជ ⌺兲 − 兺 ¯␸共⌺in,− ␷ជ ⌺兲. ⌺

⌺

共12兲

⬁

¯␸共rជ, ␷ជ 兲 = 兺 ␸共t兲p共rជ, ␷ជ ,t兲.

共5兲

t=0

For example, we can define the average exit time with simply ␸ = t

Indeed, the first sum over the surface includes all the terms which are in the first sum over the volume, but not in the second, whereas it is the opposite for the second sum over the surface. This equation slightly simplifies

⬁

¯t共rជ, ␷ជ 兲 = 兺 tp共rជ, ␷ជ ,t兲.

兺

共6兲

rជ苸V,␷ជ

t=0

¯␸共rជ − ␷ជ , ␷ជ 兲 −

兺

rជ苸V,␷ជ

¯␸共rជ, ␷ជ 兲 = 兺 ¯␸共⌺out, ␷ជ ⌺兲 − S␸共0兲. ⌺

Another such average which will be useful is the Laplace transform ⬁

pˆ共rជ, ␷ជ ,s兲 = 兺 e p共rជ, ␷ជ ,t兲 = e . −st

−st

共7兲

t=0

We will then define two spatial averages of ¯␸: the first one is the surface average 1 共8兲 兺 ¯␸共⌺out, ␷ជ⌺兲. S ⌺ In particular, 具t典⌺ is the mean time needed to return to the surface, or, alternatively, the mean number of 共not necessarily different兲 sites visited between entrance and exit. Note that the first site, which is out of the volume, is not counted, due to the definition of the probabilities. S is the surface, or the number of lines crossing the boundary. There is a second useful average which may be defined, the volume average 具␸典⌺ =

具␸典V =

1 兺 ¯␸共rជ, ␷ជ兲. V␴D rជ苸V,␷ជ

共13兲 Finally, we can easily extend the model to the case where the surface portions may be either absorbing or reflecting. For example, it may be useful to compute the mean return time to a site, if the boundary conditions are totally reflexive 共see Fig. 2兲. We denote ⌺ the points of the absorbing surface and ⌺⬘ the points of the reflexive surface, defined by the reflective boundary condition

⬘ ,− ␷ជ ⌺⬘兲 = p共rជ⬘, ␷ជ ⬘,t兩⌺out ⬘ , ␷ជ ⌺⬘兲. p共rជ⬘, ␷ជ ⬘,t兩⌺in Equation 共12兲 is thus modified the following way:

兺

rជ苸V,␷ជ

¯␸共rជ − ␷ជ , ␷ជ 兲 −

兺

rជ苸V,␷ជ

¯␸共rជ, ␷ជ 兲

⬘ , ␷ជ ⌺⬘兲 = 兺 ¯␸共⌺out, ␷ជ ⌺兲 − 兺 ¯␸共⌺in,− ␷ជ ⌺兲 + 兺 ¯␸共⌺out ⌺

共9兲

V is the volume, i.e., the number of sites inside the boundary, and ␴D is the coordination number of the lattice. For a hy-

共14兲

⌺

⌺⬘

⬘ ,− ␷ជ ⌺⬘兲. − 兺 ¯␸共⌺in ⌺⬘

The two last terms are exactly equal, and thus we have

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PHYSICAL REVIEW E 72, 016127 共2005兲

When we sum over all rជ and ␷ជ , the two last terms cancel out

兺 ¯␸共rជ, ␷ជ⬙兲 − ¯␸共rជ, ␷ជ兲 = 0

共20兲

␷ជ ⬙,␷ជ

and we can make use of Eq. 共12兲 to obtain es

冋兺 ⌺

册

pˆ共⌺out, ␷ជ ⌺,s兲 − 兺 pˆ共⌺in,− ␷ជ ⌺,s兲 = 共1 − es兲 兺 pˆ共rជ, ␷ជ ,s兲. ⌺

rជ,␷ជ

共21兲 Thus, we have S共具e−st典⌺ − 1兲 = 共e−s − 1兲␴DV具e−st典V .

共22兲

If we develop each side in powers of s, we obtain ⬁

S兺

k=1

⬁

⬁

sk共− 1兲k k sl共− 1兲l sm共− 1兲m m 具t 典⌺ = ␴DV 兺 具t 典V . 兺 k! l! m=0 m! l=1 共23兲

FIG. 2. The surface is totally reflecting; to compute the average return time to a site, we compute the average return time to the surface surrounding the site, and add 1.

兺

rជ苸V,␷ជ

¯␸共rជ − ␷ជ , ␷ជ 兲 −

兺

rជ苸V,␷ជ

¯␸共rជ, ␷ជ 兲 = 兺 ¯␸共⌺out, ␷ជ ⌺兲 − S␸共0兲, ⌺

共16兲 where S is the surface of the absorbing portion. Thus, in the sequel of the paper, we have to keep in mind that all the results apply to a lattice where a part of the boundary is reflecting. III. FIRST-EXIT TIME

We can now proceed to the computation of the moments of the first-exit time. We have the following equation for the conditional probabilities:

If we identify, we have k

1 共− 1兲k k ␴ DV 具tk−l典V . 具t 典⌺ = 兺 k! S l=1 l!共k − l兲! Finally we have the relation n

⬙

共17兲 This equation is simply the translation in terms of conditional probabilities of the rules for the behavior of the random walker. We may at once sum over all the rជ⬘ and ␷ជ ⬘ on the boundary, as indicated in Eq. 共4兲, and we have p共rជ − ␷ជ , ␷ជ ,t + 1兲 = p共rជ, ␷ជ ,t兲 +

␭ 兺 关p共rជ, ␷ជ⬙,t兲 − p共rជ, ␷ជ,t兲兴. ␴D ␷ជ ⬙

共18兲 We may now use the Laplace transforms es pˆ共rជ − ␷ជ , ␷ជ ,s兲 = pˆ共rជ, ␷ជ ,s兲 +

␭ 兺 关pˆ共rជ, ␷ជ⬙,s兲 − pˆ共rជ, ␷ជ,s兲兴. ␴D ␷ជ ⬙

共19兲

共25兲

It is possible to obtain this expression directly from the evolution equation of tn. However, the computation is slightly longer, and we will not detail it here. An important consequence of this relation is that, in the special case of the first moment, we have the following simple result: 具t典⌺ =

␭ 兺 关p共rជ⬘, ␷ជ⬘,t兩rជ, ␷ជ⬙兲 − p共rជ⬘, ␷ជ⬘,t兩rជ, ␷ជ兲兴. ␴D ␷ជ

冉冊

n ␴ DV 具t 典⌺ = 兺 具tn−m典V . S m=1 m n

p共rជ⬘, ␷ជ ⬘,t + 1兩rជ − ␷ជ , ␷ជ 兲 = p共rជ⬘, ␷ជ ⬘,t兩rជ, ␷ជ 兲 +

共24兲

␴ DV . S

共26兲

This explicit result is quite similar to the result obtained in the continuous case 关16兴: It has the same dependence on the surface and volume, but the numerical prefactor is modified. The simplicity of this equation makes it very easy to use. For instance, in the case of Fig. 2, we obtain the result that the average return time is simply V. It depends neither on the frequency ␭ nor on the shape of the volume. However, we must note that, for the higher-order moments, the results obtained are different from what we have in the continuous case 关18兴: In the latter case, the nth moment of the surfacic first-exit time depends only on the 共n − 1兲th moment of the volumic first-exit time, whereas, on a lattice, we must take into account all the 共n − 1兲 first moments of the volumic first-exit time. Furthermore, we can obtain a lower bound for 具t典V, if we inject the value of 具t典⌺ into the equation for the second moment

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CONDAMIN, BÉNICHOU, AND MOREAU

具t2典⌺ =

冉

冊

2 ␴ DV 1 具t典V + . 2 S

Since we have 具t2典⌺ 艌 共具t典⌺兲2 we finally obtain the following bound:

共28兲

␴ DV 1 共29兲 − . 2 2S Note that there may be equality only in the case when every trajectory from surface to surface has the same length. For a square lattice, this is only the case for a ballistic motion 共i.e., ␭ = 0兲 when the volume has a square shape. 具t典V 艌

IV. RESIDENCE TIME IN A SUBVOLUME

In many problems, the interesting quantity is not the time spent in the whole volume, but rather the time spent in a subpart of this volume. For instance, the random walker may be a chemical reactant which may react exclusively on several catalytic sites. Then, the subvolume corresponds to the ensemble of these catalytic sites. Here we will only derive the first moment of the residence time in this subvolume V⬘, since the other moments are of limited interest, and have a much more complicated expression. What we will compute is the average residence time in V⬘, assuming the random walk starts and finishes at the boundary of V. Thus, here, we will slightly modify the definitions of p共rជ , ␷ជ , t兲 and t共rជ , ␷ជ 兲. The time will be the time spent in the subvolume V⬘. We thus have ␭ p共rជ − ␷ជ , ␷ជ ,t + 1兲 = p共rជ, ␷ជ ,t兲 + 兺 关p共rជ, ␷ជ⬘,t兲 − p共rជ, ␷ជ,t兲兴 ␴D ␷ជ ⬘

共30兲

if r 苸 V⬘.

具t典⌺ =

共27兲

V. NUMBER OF HITS ON A REFLECTING SURFACE

Another question which may be asked is the following: On average, how many times does a particle touch a portion ⌺⬙ of the reflecting surface before exiting? If we note t the number of times the particle touches the surface ⌺⬙ before exiting, we have the following equations: 共we note in this section ⌺⬘ the rest of the reflecting surface, and ⌺ is still the absorbing surface兲 ␭ 兺 关p共rជ, ␷ជ⬘,t兲 − p共rជ, ␷ជ,t兲兴 ␴D ␷ជ

p共rជ − ␷ជ , ␷ជ ,t兲 = p共rជ, ␷ជ ,t兲 +

⬘

共36兲 and the boundary conditions p共⌺in,− ␷ជ ⌺,t兲 = ␦共t,0兲,

共37兲

⬘ ,− ␷ជ ⌺⬘,t兲 = p共⌺out ⬘ , ␷ជ ⌺⬘,t兲, p共⌺in

共38兲

⬙ ,− ␷ជ ⌺⬙,t + 1兲 = p共⌺out ⬙ , ␷ជ ⌺⬙,t兲. p共⌺in

共39兲

Using the Laplace transforms, we get the following equations: ␭ 兺 关pˆ共rជ, ␷ជ⬘,s兲 − pˆ共rជ, ␷ជ,s兲兴, ␴D ␷ជ ⬘

␭ 兺 关p共rជ, ␷ជ⬘,t兲 − p共rជ, ␷ជ,t兲兴 ␴D ␷ជ

共40兲

⬘

共31兲 if r 苸 V⬘. Thus, we have ¯t共rជ − ␷ជ , ␷ជ 兲 − 1 = ¯t共rជ, ␷ជ 兲 +

␭ 兺 关t¯共rជ, ␷ជ⬘兲 − ¯t共rជ, ␷ជ兲兴 ␴D ␷ជ

共32兲

⬘

␭ 兺 关t¯共rជ, ␷ជ⬘兲 − ¯t共rជ, ␷ជ兲兴 ␴D ␷ជ

共33兲

⬘

␭

␷,r

␷,r

rជ,␷ជ ,␷ជ ⬘

共41兲

⬘ ,− ␷ជ ⌺⬘,s兲 = pˆ共⌺out ⬘ , ␷ជ ⌺⬘,s兲, pˆ共⌺in

共42兲

⬙ ,− ␷ជ ⌺⬙,s兲 = pˆ共⌺out ⬙ , ␷ជ ⌺⬙,s兲. es pˆ共⌺in

共43兲

⬘ , ␷ជ ⌺⬘兲 兺⌺ 关pˆ共⌺out, ␷ជ⌺兲 − pˆ共⌺in,− ␷ជ⌺兲兴 + 兺 关pˆ共⌺out ⌺⬘

⬘ ,− ␷ជ ⌺⬘兲兴 + 兺 关pˆ共⌺out ⬙ , ␷ជ ⌺⬙兲 − pˆ共⌺in ⬙ ,− ␷ជ ⌺⬙兲兴 = 0. − pˆ共⌺in

if r 苸 V⬘. We sum all this, which leads to ¯t共rជ − ␷ជ , ␷ជ 兲 = 兺 ¯t共rជ, ␷ជ 兲 + 兺 兺 ␴D ជជ ជជ

pˆ共⌺in,− ␷ជ ⌺,s兲 = 1,

Here we can sum the relation 共40兲 over all rជ and ␷ជ , and use the relations 共12兲 and 共20兲, which yields

if r 苸 V⬘. ¯t共rជ − ␷ជ , ␷ជ 兲 = ¯t共rជ, ␷ជ 兲 +

共35兲

Thus, the residence time in a subvolume is proportional to its volume, and does not depend of the shape of the subvolume. We can also say that each site is visited in average ␴D / S times. We check that, if the subvolume is in fact the whole volume, we find the same result as previously 关see Eq. 共26兲兴.

pˆ共rជ − ␷ជ , ␷ជ ,s兲 = pˆ共rជ, ␷ជ ,s兲 +

p共rជ − ␷ជ , ␷ជ ,t兲 = p共rជ, ␷ជ ,t兲 +

␴ DV ⬘ . S

⌺⬙

关t¯共rជ, ␷ជ ⬘兲 − ¯t共rជ, ␷ជ 兲兴 + ␴DV⬘ .

共44兲 We define

共34兲 Equations 共16兲 and 共20兲 still apply here, and we have 016127-4

具 ␸ 典 ⌺⬙ =

1 ⬙ , ␷ជ ⌺⬙兲. 兺 ¯␸共⌺out S⬙ ⌺ ⬙

共45兲

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Thus, the law is

Using the boundary conditions, we get S共具e−st典⌺ − 1兲 + S⬙共1 − e−s兲具e−st典⌺⬙ = 0.

共46兲

If we develop this expression into powers of s, and identify, we finally get the following relation for the moments of t: n

冉冊

n n−m S⬙ 具t 典⌺⬙ . 具t 典⌺ = 兺 S m=1 m n

共47兲

We may notice that, for the first moment, we get the simple result S⬙ 具t典⌺ = . S

共48兲

This relation is again quite simple, and may easily be directly useful. VI. CASE OF A NONUNIFORM ENERGY LANDSCAPE

The most simple models of random walks, which we have considered until now, assume that all the points of the lattice are equivalent. But, in some applications, we have to take into account the fact that the vertices of the lattice may be at different potentials. For instance, in the case of a particle 共vacancy, adatom兲 moving on a crystalline lattice, the presence of an inhomogeneity 共typically, another kind of atom, which would be adsorbed in the surface兲 may modify the effective potential for our random walker around it. In the model we will introduce in this section, the various points of the lattice may have different energies. To take this particularity into account, we add to our model a reflexion probability. When going from the site x to the site y, the random walker has a certain probability Rx→y to be reflected, and a probability Tx→y to be transmitted. The probabilities satisfy a detailed balance relation, which is

冦 冦

rជ共t + 1兲 = rជ共t兲 + ␷ជ 共t兲

␷ជ 共t + 1兲 =

␷ជ 共t + 1兲 =

冦 冦

再

− ␷ជ 共t兲

prob. 1 − ␭

random prob. ␭

冎冧

prob. Rrជ共t兲→rជ共t兲+␷ជ 共t兲 . 共50兲

⌺out

具t典⌺ =

兺⌺ e−E

⌺out

共51兲

.

It is this average residence time we will compute here. We have p共rជ − ␷ជ , ␷ជ ,t + 1兲

冉

= Trជ−␷ជ →rជ p共rជ, ␷ជ ,t兲 +

冉

␭ 兺 关p共rជ, ␷ជ⬘,t兲 − p共rជ, ␷ជ,t兲兴 ␴D ␷ជ ⬘

冊

+ 共1 − Trជ−␷ជ →rជ兲 p共rជ − ␷ជ ,− ␷ជ ,t兲 ␭ 兺 关p共rជ − ␷ជ, ␷ជ⬘,t兲 − p共rជ − ␷ជ,− ␷ជ,t兲兴 ␴D ␷ជ ⬘

冊

共52兲

if rជ − ␷ជ 苸 V. As for the boundary conditions, we have

冉

⬘

冊

=R⌺out→⌺in

冉

␭ p共⌺in,− ␷ជ ⌺,t + 1兲 =R⌺in→⌺out p共⌺in, ␷ជ ⌺,t兲+ ␴ 兺 关p共⌺in, ␷ជ ⬘,t兲 − p共⌺in, ␷ជ ⌺,t兲兴 D ␷ជ p共⌺in,− ␷ជ ⌺,0兲

prob. Trជ共t兲→rជ共t兲+␷ជ 共t兲 ,

兺⌺ e−E ¯t共⌺out, ␷ជ⌺兲

␭ p共⌺out, ␷ជ ⌺,t + 1兲 =T⌺out→⌺in p共⌺in, ␷ជ ⌺,t兲+ ␴ 兺 关p共⌺in, ␷ជ ⬘,t兲 − p共⌺in, ␷ជ ⌺,t兲兴 D ␷ជ p共⌺out, ␷ជ ⌺,0兲

冎冧

As for the boundary conditions, there may be reflections on the entrance of the volume. We note A⌺out,⌺in = A⌺. Of course, if the particle is immediately reflected, the total time it will have spent inside the volume will be 0. We also redefine the average residence time: It will be the residence time weighted by the Boltzmann factors of the entry sites

共49兲

共We consider kT = 1, and scale the energies accordingly.兲

prob. 1 − ␭ ␷ជ 共t兲 random prob. ␭

rជ共t + 1兲 = rជ共t兲

+ Tx→ye−Ex = Ty→xe−Ey = Ax,y .

再

⬘

=T⌺in→⌺out .

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冊

冧

共53兲

共54兲

PHYSICAL REVIEW E 72, 016127 共2005兲

CONDAMIN, BÉNICHOU, AND MOREAU

Then, from this model, it can be shown 共see Appendix A兲 that

¯t共rជ − ␷ជ , ␷ជ ,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲 = 共1 − ␭兲2¯t共rជ, ␷ជ ,rជ⬘, ␷ជ ⬘兲 +

具t典⌺ = ␴D

兺ជ e−E

rជ

r

兺⌺ e

−E⌺

.

共55兲

␭ 共1 − ␭兲 兺 ¯t共rជ, ␷ជ ⬙,rជ⬘, ␷ជ ⬘兲 ␴D ␷ជ ⬙

+

␭ 共1 − ␭兲 兺 ¯t共rជ, ␷ជ ,rជ⬘, ␷ជ ⵮兲 ␴D ␷ជ

+

␭ 兺 ¯t共rជ, ␷ជ⬙,rជ⬘, ␷ជ⵮兲 + ␦共rជ,rជ⬘兲. ␴D2 ␷ជ ⬙,␷ជ ⵮

⵮

out

2

This relation may quite easily be understood intuitively: If the energies in the volume are lower than the energies in the surface, the random walker will have more difficulty exiting the volume, and, thus, will stay longer inside the volume: The sum 兺rជe−Erជ will increase. Inversly, if the energies in the volume are higher than the energies in the surface, then the particles will tend to be reflected immediately, and the average time spent in the volume will decrease drastically. Moreover, if the energy landscape is flat, the energies are identical everywhere in the volume and the surface, we go back to the simple result 共26兲.

共58兲

关The ␦共rជ , rជ⬘兲 is the Kronecker delta function, whose value 1 if rជ = rជ⬘ and 0 otherwise.兴 If we sum this over all rជ , rជ⬘ , ␷ជ , ␷ជ ⬘, we notice that the right-hand side of this equation nicely simplifies

兺

rជ,rជ⬘,␷ជ ,␷ជ ⬘

¯t共rជ − ␷ជ , ␷ជ ,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲 =

兺

rជ,rជ⬘,␷ជ ,␷ជ ⬘

2 ¯t共rជ, ␷ជ ,rជ⬘, ␷ជ ⬘兲 + ␴D V.

共59兲 VII. JOINT RESIDENCE TIME FOR TWO PARTICLES

We will now consider not one, but several particles diffusing independently, and we wish to compute joint residence times. This is interesting in the case where we have several particles which must be on the same site to react 关20兴. For example, it may be two chemical reactants, or a vacancy and an adatom. If the two particles are strongly interacting, they will react as soon as they are in contact. The interesting quantity here is the meeting probability P. We cannot compute it directly, but, given the average interaction time, we may have an upper bound P 艋 具t典.

共56兲

The last term is simply the number of quadruplets 共rជ , rជ⬘ , ␷ជ , ␷ជ ⬘兲 where rជ = rជ⬘. We have the following relations, similar to Eq. 共16兲: ¯t共rជ − ␷ជ , ␷ជ ,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲 兺 ជជ r,␷

= 兺 ¯t共rជ, ␷ជ ,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲 + 兺 ¯t共⌺out, ␷ជ ⌺,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲 rជ,␷ជ

− 兺 ¯t共⌺in,− ␷ជ ⌺,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲.

The last term in this equation is zero, since if a particle is at the position ⌺in with the speed −␷ជ ⌺, it immediately exits. We thus have

兺

rជ,rជ⬘,␷ជ ,␷ជ ⬘

=

Thus, it is always interesting to compute the joint residence time. So, we consider here two particles, which may either start from the bulk or the boundary of the volume. We define the joint residence time as the amount of time spent by the two particles on the same site before one of them exits. Thus, we can define q共rជ , ␷ជ , rជ⬘ , ␷ជ ⬘ , t兲, the probability that the two particles meet t times before they exit, given that the first one starts from the position rជ with the speed ␷ជ , and the second one starts from the position rជ⬘ with the speed ␷ជ ⬘. We can also define the average interaction time given the initial positions and speeds ¯t共rជ , ␷ជ , rជ⬘ , ␷ជ ⬘兲. We have the following equation:

¯t共rជ − ␷ជ , ␷ជ ,rជ⬘ − ␷ជ ⬘, ␷ជ ⬘兲

兺

rជ,rជ⬘,␷ជ ,␷ជ ⬘

+ 共57兲

共60兲

⌺

共It is interesting, since the average interaction time will genraly be much lower than 1.兲 Otherwise, if the two particles are weakly interacting 共i.e., have a small probability p of reacting each time they meet兲, then the reaction probability will be approximatively P = p具t典.

⌺

¯t共rជ, ␷ជ ,rជ⬘, ␷ជ ⬘兲 +

兺

⌺,␷ជ ⬘,rជ⬘

¯t共⌺out, ␷ជ ⌺,rជ⬘, ␷ជ ⬘兲

⬘ , ␷ជ ⌺⬘ 兲 + 兺 ¯t共⌺out, ␷ជ ⌺,⌺out ⬘ , ␷ជ ⌺⬘ 兲. 兺 ¯t共rជ, ␷ជ,⌺out

rជ,␷ជ ,⌺⬘

共61兲

⌺,⌺⬘

Finally, reporting this result in Eq. 共59兲, we have 2

兺

⌺,␷ជ ⬘,rជ⬘

¯t共⌺out, ␷ជ ⌺,rជ⬘, ␷ជ ⬘兲 +

⬘ , ␷ជ ⌺⬘ 兲 = ␴D2 V. 兺 ¯t共⌺out, ␷ជ⌺,⌺out

⌺,⌺⬘

共62兲 If we define the following average joint occupation times: 具t典⌺V =

1 兺 ¯t共⌺out, ␷ជ⌺,rជ⬘, ␷ជ⬘兲 V␴DS ⌺,␷ជ ,rជ

共63兲

⬘ ⬘

is the average joint occupation time when a particle starts from the volume and the other from the surface, and

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FIRST-EXIT TIMES AND RESIDENCE TIMES FOR …

具t典⌺2 =

1 ⬘ , ␷ជ ⌺⬘ 兲 兺 ¯t共⌺out, ␷ជ⌺,⌺out S2 ⌺,⌺

PHYSICAL REVIEW E 72, 016127 共2005兲

for higher-order moments 共25兲兴. It should be pointed out that we obtained explicit exact results 关Eqs. 共26兲, 共35兲, 共48兲, and 共55兲兴, which are not so common in the bounded random walks literature. Furthermore, they apply to all kinds of lattices and random walks: Because they have a very general range of application, these results can be very useful when the evolution equations cannot be solved exactly. In fact, it is even possible to generalize our methods to irregular networks, where the connectivity can vary from one site to another. This may be of special interest for complex networks which can be found in ethology, economy, neural networks, or social sciences. Such an extension is in progress.

共64兲

⬘

is the average joint occupation time when the two particles start from the surface. Then, we have the following result:

␴D2 V = 2␴DVS具t典⌺V + S2具t典⌺2 .

共65兲

Thus, we cannot have a totally explicit result for the joint residence time; we just derived a relation between two different joint residence times, depending on where the two particles start from. However, this relation gives immediately a useful upper bound to these joint residence times

␴2 V 具t典⌺2 艋 D2 , S

共66兲

␴D . 2S

共67兲

具t典⌺V 艋

APPENDIX A: COMPUTATION OF THE MODEL WITH ENERGIES

We can rewrite Eqs. 共52兲, 共53兲, and 共54兲 in terms of average time ¯t共rជ − ␷ជ , ␷ជ 兲 − 1

Since, in trial cases, we found out that the magnitude of the two terms in the relation 共65兲 was similar, we have good hope that these relations will at least give an upper bound which is of the good order of magnitude. It is possible to have a similar result for n particles: It is a relation between n different averages 共depending on the number of particles which start from the boundary, and the number of particles which start from the volume. The result 共see Appendix B for details on the computation兲 is the following: n

␴DnV

=

兺 m=1

冉冊

n m S 共␴DV兲n−m具t典⌺mVn−m , m

冉

⫻ ¯t共rជ − ␷ជ ,− ␷ជ 兲 +

共68兲

r,␷,r−␷苸V

r,␷,r−␷苸V

+

兺 ជជជ ជ

冉

r,␷,r−␷苸V

冉

⬘

冊

␭ 兺 关t共⌺in, ␷ជ⬘兲 − t共⌺in, ␷ជ⌺兲兴 + 1. ␴D ␷ជ ⬘

共A2兲 Then we have t共⌺out, ␷ជ ⌺兲 = T⌺out→⌺in␣⌺ ,

共A3兲

t共⌺in,− ␷ជ ⌺兲 = 共1 − T⌺in→⌺out兲␣⌺ .

共A4兲

If we sum the relation 共A1兲 over all rជ and ␷ជ , weighted by the appropriate Boltzmann factors, we get

Arជ,rជ−␷ជ ¯t共rជ, ␷ជ 兲 − ¯t共rជ − ␷ជ ,− ␷ជ 兲 e−Erជ−␷ជ ¯t共rជ − ␷ជ ,− ␷ជ 兲 +

␭ 兺 关t¯共rជ − ␷ជ, ␷ជ⬘兲 − ¯t共rជ − ␷ជ,− ␷ជ兲兴 ␴D ␷ជ

␣⌺ = t共⌺in, ␷ជ ⌺兲 +

The results obtained in this paper significantly extend the results previously derived 关16,18,19兴. They show that the mean first-exit time behaves differently for a discrete lattice and for a continuous media, not only quantitatively 关as for Eq. 共26兲, for instance兴, but also qualitatively 关see the relation

兺 ជជជ ជ

⬘

if rជ − ␷ជ 苸 V, if we define

VIII. CONCLUSION

e−Erជ−␷ជ关t¯共rជ − ␷ជ , ␷ជ 兲 − 1兴 =

冊

␭ 兺 关t¯共rជ, ␷ជ⬘兲 − ¯t共rជ, ␷ជ兲兴 + 共1 − Trជ−␷ជ→rជ兲 ␴D ␷ជ

共A1兲

where 具t典⌺mVn−m is the average joint residence time for n particles, of which m start from the surface and n − m from the boundary.

兺 ជជជ ជ

冉

= Trជ−␷ជ →rជ ¯t共rជ, ␷ជ 兲 +

␭ 兺 关¯t共rជ, ␷ជ⬘兲 − ¯t共rជ − ␷ជ, ␷ជ⬘兲+ ¯t共rជ − ␷ជ,− ␷ជ兲 − ¯t共rជ, ␷ជ兲 兴 ␴D ␷ជ ⬘

冊

␭ 兺 关t¯共rជ − ␷ជ, ␷ជ⬘兲 − ¯t共rជ − ␷ជ,− ␷ជ兲兴 . ␴D ␷ជ ⬘

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PHYSICAL REVIEW E 72, 016127 共2005兲

CONDAMIN, BÉNICHOU, AND MOREAU

We will see how all these terms may transform.

兺 ជជជ ជ

r,␷,r−␷苸V

e−Erជ−␷ជ¯t共rជ − ␷ជ , ␷ជ 兲 = 兺 e−Erជ¯t共rជ, ␷ជ 兲 − 兺 e−E⌺in¯t共⌺in,− ␷ជ ⌺兲. rជ,␷ជ

We use the relation 共A4兲 on the left-hand side, and recognize ␣⌺ on the right-hand side, we get

兺⌺ 共− e−E

⌺

共A6兲

兺

rជ,␷ជ

−E⌺

in

⌺

Arជ,rជ−␷ជ¯t共rជ, ␷ជ 兲 = 兺 Arជ,rជ−␷ជ¯t共rជ, ␷ជ 兲 − 兺 ¯t共⌺in, ␷ជ ⌺兲, rជ,␷ជ

+ A⌺兲␣⌺ − 兺 e−Erជ + 兺 e−E⌺in

=−兺e

We also have rជ,␷ជ ,rជ−␷ជ 苸V

⌺in

共␣⌺ − 1兲.

r,␷,r−␷苸V

Arជ,rជ−␷ជ¯t共rជ − ␷ជ ,− ␷ជ 兲 =

具t典⌺ =

兺 Arជ⬘−␷ជ⬘,rជ⬘¯t共rជ⬘, ␷ជ⬘兲

rជ⬘,␷ជ ⬘

− 兺 ¯t共⌺in, ␷ជ ⌺兲.

共A8兲

⌺

r,␷,r−␷苸V

兺

Arជ,rជ−␷ជ¯t共rជ − ␷ជ , ␷ជ ⬘兲 =

⌺

兺 Arជ⬙−␷ជ⬙,rជ⬙¯t共rជ⬙, ␷ជ⬘兲 − 兺⌺ ¯t共⌺in, ␷ជ⬘兲.

rជ⬙,␷ជ ⬙

共A10兲 共We just take ␷ជ ⬙ = −␷ជ , rជ⬙ = rជ − ␷ជ .兲 These two terms also exactly cancel each other because of the detailed balance condition. We have of course similar relations with the terms in e−Erជ−␷ជ. We now use all these relations in our main equation, which yields e−E ¯t共rជ, ␷ជ 兲 − 兺 e−E 兺 ជជ ⌺ rជ

¯t共⌺in,− ␷ជ ⌺兲 − 兺 e−Erជ + 兺 e−E⌺in

⌺in

r,␷

rជ,␷ជ

⌺

兺

rជ1,␷ជ 1,…,rជn,␷ជ n

⌺

=

␭ + 兺 e−Erជ关t¯共rជ, ␷ជ⬘兲 − ¯t共rជ, ␷ជ兲兴 ␴D rជ,␷ជ ,␷ជ ␭ 兺 兺 e−E⌺in关t¯共⌺in, ␷ជ⌺兲 − ¯t共⌺in, ␷ជ⬘兲兴. ␴D ⌺ ␷ជ ,␷ជ

共A11兲

⬘

兺

rជ1,␷ជ 1,…,rជn,␷ជ n

¯t共rជ1 − ␷ជ 1, ␷ជ 1,…,rជn − ␷ជ n, ␷ជ n兲 =

兺

rជ1,␷ជ 1,…,rជn,␷ជ n n

+

兺 m=1

rជ

r

兺⌺ e−E

⌺

.

共A14兲

兺ជ ជ

n ¯t共rជ1, ␷ជ 1,…,rជn, ␷ជ n兲 + ␴D V.

共B1兲

n V is simply the number of 2n uplets Again, ␴D 共rជ1 , ␷ជ 1 , … , rជn , ␷ជ n兲 which satisfy rជ1 = ¯ = rជn. We will also have the relation, similar to the relation 共61兲, where we put together the terms which are similar

¯t共rជ1, ␷ជ 1,…,rជn, ␷ជ n兲

冉冊 n m

兺ជ e−E

¯t共rជ1 − ␷ជ 1, ␷ជ 1,…,rជn − ␷ជ n, ␷ជ n兲

rជ1,␷1,…,rn,␷ជ n

⬘

+

共A13兲

,

Here we will evaluate the joint residence time for n particles, i.e, the amount of time they will spend all in the same site. We will not write the equations in the fullest detail, since it would take pages, but the calculation is quite similar to the calculation for only two particles. We will define the times ¯t共rជ1 , ␷ជ 1 , … , rជn , ␷ជ n兲, which are the average time the n particles will spend together, given that the kth particle starts at the position rជk, with the speed ␷ជ k. These times obey a relation quite similar to Eq. 共58兲, but this relation is quite unwritable in the case of n particles. However, we can see that the right-hand term will simplify the same way it does for two particles, and we will get the equivalent of the relation 共59兲

= 兺 e−Erជ¯t共rជ, ␷ជ 兲 − 兺 e−E⌺in¯t共⌺in, ␷ជ ⌺兲 rជ,␷ជ

⌺out

APPENDIX B: COMPUTATION OF THE JOINT RESIDENCE TIMES FOR n PARTICLES

共A9兲

rជ,␷ជ ,rជ−␷ជ 苸V

兺⌺ e−E

具t典⌺ = ␴D

Arជ,rជ−␷ជ¯t共rជ, ␷ជ ⬘兲 = 兺 Arជ,rជ−␷ជ¯t共rជ, ␷ជ ⬘兲 − 兺 ¯t共⌺in, ␷ជ ⬘兲, rជ,␷ជ

兺⌺ A⌺␣⌺

we get the relation

共We just take ␷ជ ⬘ = −␷ជ , rជ⬘ = rជ − ␷ជ .兲 Note that these two expressions are identical only because of the detailed balance condition, which implies that the function A has to be symmetrical. It is important since these two terms must cancel each other. We must also compute

兺 ជជជ ជ

共A12兲

Since, because of the relation 共A3兲, we have

⌺

共A7兲

兺 ជជជ ជ

⌺

兺

兺

⌺共1兲,…,⌺共m兲 rជm+1,␷ជ m+1,…,rជn,␷ជ n

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共1兲 共1兲 共m兲 共m兲 ¯t共⌺out , ␷ជ ⌺ ,…,⌺out , ␷ជ ⌺ ,rជm+1, ␷ជ m+1,…,rជn, ␷ជ n兲.

共B2兲

FIRST-EXIT TIMES AND RESIDENCE TIMES FOR …

PHYSICAL REVIEW E 72, 016127 共2005兲

The average joint residence times are defined by 具t典⌺mVn−m =

1 共1兲 共1兲 共m兲 共m兲 , ␷ជ ⌺ ,rជm+1, ␷ជ m+1,…,rជn, ␷ជ n兲. , ␷ជ ⌺ ,…,⌺out 兺 兺 ¯t共⌺out Sm共␴DV兲n−m ⌺共1兲,…,⌺共m兲 rជm+1,␷ជ m+1,…,rជn,␷ជ n

共B3兲

We thus get the relation n

␴DnV

=

兺 m=1

冉冊

n m S 共␴DV兲n−m具t典⌺mVn−m . m

关1兴 G. Weiss, Aspects and Applications of the Random Walk 共North-Holland, Amsterdam, The Netherlands, 1994兲. 关2兴 F. Spitzer, Principles of Random Walk 共Springer-Verlag, New York, 1976兲. 关3兴 C. Itzykson and J.-M. Drouffe, Statistical Field Theory 共Cambridge University Press, Cambridge, UK, 1989兲. 关4兴 N. Goel and N. Richter-Dyn, Stochastic Models in Biology 共Academic Press, New York, 1974兲. 关5兴 Surface Dynamics, edited by D. Woodruff 共Elsevier Science B.V., 2003兲, Chap. 12. 关6兴 R. van Gastel, E. Somfai, S. B. van Albada, W. van Saarloos, and J. W. M. Frenken, Phys. Rev. Lett. 86, 1562 共2001兲. 关7兴 O. Bénichou and G. Oshanin, Phys. Rev. E 64, 020103 共2001兲. 关8兴 E. Montroll, J. Math. Phys. 10, 753 共1969兲. 关9兴 W. McCrea and F. Whipple, Proc. R. Soc. Edinburgh 60, 281 共1940兲.

关10兴 关11兴 关12兴 关13兴 关14兴 关15兴 关16兴 关17兴 关18兴 关19兴 关20兴

016127-9

共B4兲

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