PHYSICAL REVIEW B VOLUME 30, NUMBER 1 1 JULY 1984 Classical diffusion, drift, and trapping in random percolating systems R. B. Pandey" Institute of Theoretical Physics, Cologne University, 5000 Koln 41, 8'est Germany (Received 21 February 1984) Monte Carlo studies for a biased diffusion are made on simple-cubic random lattices containing 180 and 256 sites with time steps up to 10~. Above the percolation threshold, we observe diffusion for short times, and drift for long times, when the bias is below a characteristic value. For larger bias, a very slow relaxation, presumably to the asymptotic nonclassical behavior, is observed. Recently, the problem of classical diffusion on random percolating clusters has been studied intensively. ' In most of these current investigations one asks many questions: What is the power-law behavior of the root-mean-square (rms) displacement of a random walker~ How does it ap- proach its asymptotic value? How does it depend on the concentration p of the occupied sites? Earlier controversies2 between the computer experiments and scaling theories on such questions seem to be settled now for unbiased dif- fusion on isotropic lattices. 7 It is well established that, in the power-law description for the rms displacement R(t) with time t, in three dimensions, R(r)~ r". . . . The exponent k = 0 for p ( p, where R (~) is the average radius of the clusters, k = 0.20 + 0.01 at p = p„and k = 0.5 for p & p„where Eq. (1) describes Einsteinian diffusion with adiffusivity D which goes to zero for p p„p, being the percolation threshold. What happens when a biased field is switched on to cause the random walker to move with unequal probabilities (1+8)/2 and (1 — 8)/2 with 0 ( 8 & 1 in positive and negative directions, respectively? The random-walk motion diffuses the particle according to R~ Jt while the biased field causes it to drift with R~ t. When t is very small, diffusion dominates and when t is large, drift dominates. A crossover from diffusion to drift behavior thus occurs at about t, „—1/(bias field). ' Barma and Dhar have recently predicted that above the percola- tion threshold the drift velocity is nonmonotonic and van- ishes above a critical value. On the other hand, Bottger and Bryksin in a somewhat different model have shown that in high field, the current (presumably due to drift) decreases and goes to zero only in the infinite field (corresponds to 8=1 here, and in Ref. 8). To examine the controversy on biased diffusion we perform a computer simulation to study various power laws, crossover, and relaxation due to com- petition of diffusion, drift, and trapping. Our simulations seem to support the suggestions of Bottger and Bryksin. The basic idea to model the problem of biased diffusion on computer is simple. As in our previous studies7 of ordi- nary diffusion we first prepare the sample (called lattice realization) by distributing a fraction p of occupied sites ran- domly on a L xL x L simple-cubic lattice (a quenched disor- dered lattice). A particle (random walker) is then placed on a randomly selected occupied site (called local origin). The biased probability B is set up to study the motion, for which a random number r is chosen randomly between 0 and 1 and is compared with B; if r is less than or equal to 8 then an attempt is made to move it to one of the randomly 104 R 10 ~—~—0 QQ o 0.05 + 02 0.4 0.8 ~ 0.98 ~A + y4 a. ++)& o 4 ) p t~ + o 0 &e c 4 o ~ 0 a + o ~ ~ &gt p . ~ ~H bf p .& ~ ~ & 5+ pH ~ ~ yO e4 e+~ 1.0 — ~ ' + I 'l0 I 10 I 10 I 10' I 10 10 t FIG. 1. Log-Log plots of the rms displacement R with time t at concentration p=0. 5 for the bias 8=0.0, 0.05, 0.2, 0.4, 0.8, and 0.98 with their respective symbols indicated in inset. selected neighbors in the positive directions (i.e. , in +X, + Y, or + Z directions), otherwise, i. e. , if r & 8, the move is attempted to any of its six randomly selected neighbors. The bias 8 thus acts as a bias field pushing the stalker into positive directions [with probability (1+8)/2] reducing its chance to turn back [in negative direction with probability (1 — 8)/2]. Note that this study is different from the other investigations of unbiased diffusion on the biased percolat- ing clusters. ' The random walker is moved to the neigh- boring site so chosen, if the site is occupied; otherwise it stays at the same position. Each attempt is counted as one time step. The process of selecting a neighboring site under the biased probability prescription and an attempt to move it is repeated again and again until the desired (preset) time step is reached. For reliable statistics at concentration p, the simulation is performed on several independent lattice reali- zations each with many independent local origins. The average mean-square displacement is calculated over these statistics and its value is recorded at various periods of in- tervals. For our studies here we perform the simulation mainly on samples 180 on a CDC Cyber-76 machine and on 2563 samples using a CDC Cyber-205 vector machine. Apart from the biased field descriptions, the technical de- tails may be found in Ref. 7. The variation of rms displacement with time for various biased probabilities on p = 0.5 random lattices is displayed in Fig. 1. The data for R in the "small time regime" (up to 105 time steps) were generated on 1803 samples with 50 ar- 30 489 O1984 The American Physical Society