Volume 108A, number 4 PHYSICS LETTERS I April 1985

ABSENCE OF SELF-DIFFUSION IN DIRECTED PERCOLATION LATTICES

Toshiya OHTSUKI and Thomas KEYES

Department of Chemistry, Boston University, Boston, MA 02215, USA

Received 12 October 1984

Self-diffusion in directed percolation lattices is investigated theoretically. It becomes evident that perfect trapping of a particle by dead ends causes an exponential decay of a probability distribution function and vanishment of a self-diffusion coefficient except at p = 1, where p is a percolation probability. The trapping rate in d-dimensional hypercubic lattices is calculated exactly and that on the backbone of percolation clusters in a square lattice is obtained on the basis of a real-space

renormalization group method.

The problem of self-diffusion in percolating sys- tems has been of considerable current interest. Anom- alous time-dependence of the mean square displace- ment due to the fractal nature of percolation clusters has been elucidated by a number of works based upon various techniques such as scaling theory [ 1 ], real- space renormalization group (RSRG) method [2] and computer simulations [3]. Recently, the focus of in- vestigations has turned to generalization of the simple model, in particular to biased random walks, because of appearance of completely new features and realiza- tion in a wide variety of real physical systems [4 -9 ] . Ohtsuki [4] Bottger and Bryksin [5] and Barma and Dahr [6] suggested that a strong bias (external field) causes trapping of particles in dead ends and decrease of mobility. On the basis of the RSRG method, Ohtsuki and Keyes [7] derived a criterion for the trapping and clarified that near a percolation thresholdp c, the critical bias is in inverse proportion to the correlation length of systems. On the other hand, directed percolation has also been the subject of intensive studies [ 10]. From a viewpoint of biased diffusion, a directed system corre- sponds to the limiting case of strong bias. In this limit, a particle can move only in some preferred directions and migration in opposite directions is inhibited. Then the previous arguments indicate that the particle is cap- tured perfectly and the self-diffusion coefficient becomes zero except at p = 1, where p is the percolation prob- ability [6]. In other words, the percolation threshold for self-diffusion is unity at all dimensions. The pur-

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pose of this letter is to investigate self-diffusion in di- rected percolation lattices and to clarify the effects of the trapping.

We consider a directed random walk in a 2-ddmen- sional square lattice. We treat bond percolation where bonds are connected with a probability p and broken with a probability q = 1 - p. A rule for elementary particle motions is given as follows (cf. fig. 1). There are two preferred directions in which particles can move. When both bonds in these directions are con- nected, a particle walks one of them with an equal probability 1/2 (fig. la). If one of the preferred bonds is broken and the other is connected, a particle jumps the connected bond with a probability unity (fig. lb). Lastly, when both bonds are broken, a particle can- not move any more and is captured perfectly (fig. lc). Each process occurs with a probability p2, 2pq and q2, respectively. A particle starting at the origin (i ,]) =

t~z o o I I I I I i

I 1 I - ?- 0 ~ ~ 0 • . . . . . 0

¢ ) (b) (c)

Fig. 1. Elementary processes of a directed random walk in a 2-dimensional square lattice with connected ( - - ) and broken ( - - - ) bonds.

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(0, 0) travels parallel to the anisotropy axis described by i = / and is trapped with a finite probability. If the particle remains free after N walks, it reaches sites (i , /) on a line i + j = N ( i , j >1 0). It should be emphasized that each step is completely independent, because a particle never returns to the sites where it visited before and never passes the same bond more than once. Hence, the trapping probability at each step is always given by q2 and the distribution of free particles on the line i +/' = N follows a simple binomial distribution. As a result, the probability distribution function F = F(i, / ;A t) o f a particle a f te rN steps from the start at the origin becomes

F(i , j ; N) = (l --q2)N NC i 2 - N

(i + j = N; i , j /> 0) ; ( la)

= q2(1 _q2)M MC i 2 -M

(i +j = M ; N - 1 >~M>~ O;i,j l>O) ; ( lb)

= 0 (otherwise). ( lc )

Here the term ( la) represents the distribution of free particles on the line i + j = N and the term ( lb) stands for that of trapped particles at site (i,j). The probabil- ity P o f being free decays exponentially

P(N) = (1 _ q 2 ) N = e x p ( - k N ) , (2)

with a rate constant

k = - ln(1 _ q 2 ) . (3)

In the long time limit N ~ 0% therefore, the mean dis- placement ((R - (R))2) of the particle approaches asymptotically a finite value and the self-diffusion co- efficient D s defined by D s - ((R - (R))2)/2N goes to zero. Similarly to the case o f diffusion among static traps, the distribution function of free particles obeys macroscopically a reaction-diffusion equation (with a drift term due to the deterministic directed motion). When q ~ 1, k ~- q2. This means that at least two broken bonds are necessary to form a trap.

Eqs. (1 ) - (3 ) are exact results for any p under the condition that a starting point (origin) o f diffusion is chosen randomly with an equal probability for all sites. Just above the percolation threshold Pc, however, the system consists of one infinite cluster and many Finite clusters and the probability o f belonging to the infinite cluster is infinitesimally small. The trapping rate in the infinite or large clusters is obviously smaller than

that in small clusters. Especially, if a particle is laid in the backbone of the infinite duster, there exists a path o f infinite extent where a particle can travel. Thus, the survival probability PBB of a particle put in the backbone may decay more slowly than that given by eqs. (2) and (3). Here we calculate PBB on the basis of the RSRG technique.

A cell-to-cell decimation scheme is adopted and a group of sites and bonds forming a triangular cell o f edge m are combined into supersites and superbonds in a renormalized cell o f edge m'. The example o f m = 2 and m' = 1 is shown in fig. 2. In order to take into account the effects of the trapping, we assume that a particle is captured even by connected bonds and intro- duce the permeability a of a connected bond which represents a survival probability during the walk on the bond. Then the survival probability of a particle starting on the lower left corner of the cell until it reaches sites on the upper right base line is computed for each configuration of the cell with respect to con- nected and broken bonds. Next, we have to take an average over cell configurations. Here the appropriate procedure is to average only over connected configura- tions where the lower left corner is connected to the upper right base line, because of the existence of a path o f infinite extent (backbone). In general, the averaged probability has the form ofg(p, m) a m . The survival probability g(p' , m ' ) a 'm ' in the renormalized cell is similarly obtained, where primes denote renor- malized quantities. In the spirit of the RSRG approach, macroscopic properties of systems are kept invariant under the transformation. In this case, a relevant quantity is the survival probability. Putting g(p' , m ' ) × ct 'm' =g(p, m)ct m , therefpre, we have a recursion rela- tion for ct

a'(a') = f ( p , p ' ; m, m') [a(a)] b , (4)

. O :::~

Fig. 2. Cell-to-cell decimation scheme with m -- 2 and m' = 1.

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where b = m/m' is a rescaling factor and a is the lat- tice constant. Since no trapping occurs at connected bonds in the original lattice, a(1) = 1, where the lat- tice constant of the original lattice is taken as unity. Iterating the transformation n times, we get

n - 1 o~(N) = I-I [ l~i ,Pi+l;m,m')]bn- l - i , (5)

i=0

where N = b n and Pi is the percolation probability after i iterations.

At the percolation threshold p = Pc, Pi = Pc for all i and eq. (5) gives

eBB(N) = ~ N ) = bee(Pc; m, m')] (bn-1)](b-1)

= exp [--kBB(N-- 1)1 , (6)

kBB = -- [ln re(Pc; m, m ' ) l / ( b - 1). (7)

It should be noted that even in the presence of an in- finite path (backbone), the survival probability PBB decays exponentially, because a particle is trapped perfectly by dead ends with a finite probability. How- ever, the rate constant kBB is smaller than that (k) for random initial location. Explicit calculations are per- formed for m = 2 and m ' = 1. The result is

f (p ,p ' ;2 , 1 ) = f ( p ; 2, 1)

p2 + 2pq q2 = 1 - . ( 8 )

p4 +6p3q + 14pZq2 + 14pq3+ 4q4

Using a known estimate p e ~ 0.6446 [ 1 1 - 1 4 ] , we evaluate kBB as

kBB ~ 0.058. (9)

This value is about a half o f that k = ln(1 _ q 2 ) ~-0.135 given by eq. (3). Note that if the average is taken over all configurations in calculations o f g , f is given by (1 --q2)b/(1 _ q ' 2 ) . Since P(N) = (1 - q 2 ) a ( N ) in this case, the present scheme reproduces the exact results (2) and (3) for all p. Furthermore, eq. (8) reduces to the exact value 1 _ q 2 in the limit p ~ 1.

In the vicinity of a percolation threshold as e - (t 7 -Pc)/Pc ~ 1,Pi =Pc + O(e) at b i ,~ ~11, where ~11 is a longitudinal correlation length parallel to the ani- sotropy axis [10], whereas Pi ~' 1 when b i >> ~11" Since f ~ " 1 a t p ~" 1,PBB(N) = ct(N) is dominated byf(pi,

~, • m "~ ~11 and is given Pi+l ; m, m') fc(Pc, , m') when b i by

PBB(N) ~ Ire(Pc; m, m')] (bn- 1)/(b-l)

= e x p [ - - k B B ( N - 1)1 ( N ~ l l ) ;

[fe(Pc; m, m') ] (bn-bn- 1-/)/(b- 1)

= exp[--kBB(N-N[~II)] (N>> ~11),(10)

where b / = ~tl" In the case ~11>~ 1, therefore, the decay (10) Of PBB becomes almost the same as that (6) at P = Pc" In addition, the self-similarity of percolation clusters informs us that even when a particle starts on dead ends or finite clusters, eqs. (6) and (7) hold until N approaches to the size L of dead ends or finite clus- ters in the direction of the anisotropy axis from the starting point to the end of the structures. Then the rate constant kBB is considered to play an important role in various transport properties of directed percola- ting systems near the percolation threshold.

The extension of the preceding arguments to higher dimensional systems is straightforward. In d-dimen- sional hypercubic lattices, for example, a particle can walk in d preferred directions and the distribution function F d( i l , i 2 ..... i d ; N ) becomes

Fd(i l , i 2 ..... id;N)

= (1 --qd)NN! il! d -N ( l l a )

(Zd=l i t = N; i l >/O) ;

d =qd(1--qd)M M, (l~=l il' ) - l d -M ( l l b )

(Ed=l i I = M ; N - 1 >~M>~O;il>~O);

= 0 (otherwise). ( l l c )

Since a particle is captured with a probability qd in. dependently at each step, the survival probability Pd(N) follows an exponential decay

ed(N) = (1 _ q d ) N = e x p ( _ k d N ) ' (12)

k d = --ln(1 _qd) .

Redner [15] discussed conduction in partially directed percolation networks where bonds are direct- ed only in one direction and the system is isotropic in remaining directions. In this case, exact results like eqs. (1 ) - (3 ) and (11)--(13) cannot be obtained easily,

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because a particle can come back to sites visited before in isotropic directions and each step is not completely independent. At p < I , however, traps are formed and a particle is captured perfectly with a finite probabil i ty. Then the survival probabil i ty PpD(N) is considered to decay exponentially and the self-diffusion coefficient is thought to be zero. In d-dimensional hypercubic lattices, at least 2 d - 1 broken bonds are necessary for the formation of a trap. When q < 1, therefore, the rate constant kpD is given by kpD ~' q 2d-1 . We consider that as long as some preferred directions exist and particle migration in opposite directions is prohibited, self-diffusion is generally absent except a t p = 1.

In this letter, we have clarified the exponential decay of the probabil i ty distribution function and the absence of self-diffusion in directed percolation lat- tices. In contrast, a conductivity or equivalently a con- centration diffusion coefficient has a nonzero value above the percolation threshold [ 15]. It is well known that self-diffusion is not necessarily identical to con- centration diffusion [ 16]. Usually, however, the dif- ference is only quantitative. In ordinary (isotropic) percolation lattices, for instance, the self-diffusion co- efficient on the infinite percolation cluster decreases more slowly than the conductivity (concentrat ion dif- fusion coefficient) near the percolation threshold but they vanish at the same point (threshold) [1,17]. In directed lattices, as is seen here, the difference is qua-

litative. This is considered to reflect a kind of broken- ergodicity due to the inhibition of particle motions in opposite directions to preferred axes. It seems that in random systems, a strong bias or external field general- ly exerts serious influence on transport properties.

This work was part ly supported by the NSF, grant number CHE 83-12722.

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