Exact Algorithm ford-Dimensional Walks on Finite and Infinite Lattices with Traps

Abstract

An exact algorithm is formulated to calculate the expected walk length (n) for a walker (atom, molecule) undergoing random displacements on a finite or infinite (periodic) d-dimensional lattice with traps (reactive sites). The method is illustrated for the case of a single deep trap surrounded by shallow traps and the calculated value of (n) agrees to within 0.3% of the Monte Carlo result for all lattices considered. The theoryintroduced is capable of generalization to many new classes of problems in lattice statistics.PACS numbers: 05.40.+iTo investigate the interplay between spatial extent and dimensionality in influencing reactiondiffusion processes, we have undertaken recently a program of Monte Carlo calculation of random walks on lattices with traps. 1 " 3 In the course of analyzing recent data generated in these studies, we have found that a quite general algorithm can be written down to predict the number (n) of steps required before trapping (i.e., the expected walk length) on finite and infinite (i.e., periodic) lattices subject to a variety of boundary conditions. Moreover, the method developed seems capable of much further generalization so that although we consider specifically the case of a deep trap (trapping probability, T = 1) surrounded by iV-1 sites which may be partially absorbing (trapping probability, 0^ s < 1), a great variety of other situations seems accessible to analysis. The method is based on a classification of the symmetry of the sites surrounding the deep trap and a coding of the fate of the random walker as it encounters a site of given symmetry. We illustrate the method using two simple examples and cite a number of additional, representative results to show the generality of the method.Consider a 5x5 periodic lattice with a deep trap (T = 1) at the center and JV-1 = 24 surrounding sites, each characterized by a trapping probability s. From the symmetry of the unit cell, it is seen that there are five "types" of lattice sites, <n> 1 = i«n> r + l) + i«w> 2 + l)+i«w> 2 + l)+i«n) 3 + D.Since the expected walk length from a deep trap (T = 1), (n) T , is zero this expression simplifies to {n) 1 = $[l+2((n) 2 + l) + {(n) 3 + l)].

Sandpile models and random walkers on finite lattices

“…From this data, the distribution of path lengths P L (n) was generated and its average n L was calculated. The average n L was also calculated using the Walsh-Kozak method [37,38]. The distribution P L (n) was also calculated using the Soler method [39] and n L was extracted from it.…”

“…The boundaries are open, and thus a random walker starting at any site (i x , i y ) will eventually fall off the edge [37,38]. The number of moves it will make depends on the location of the initial site as well as on the particular realization of the random moves generating the path of the given walker.…”

“…The average path length of a random walker on a lattice of size L is calculated exactly using a method proposed by Walsh and Kozak [37,38]. The entire distribution of the path lengths of random walkers starting at random sites on the finite lattice is also calculated using a related method proposed by Soler [39].…”

Surface site diffusion and reaction on molecular “organizates” and colloidal catalysts: A geometrical perspective

“…Presented in Table 1 are data on the mean reaction time, as calibrated by the mean walk length (see Section 4), for a coreactant migrating on an N-site regular polyhedral lattice with a fixed metric I and a single reaction center, calculated by using the theory of finite Markov processes (12)(13)(14)(15)(16)(17)(18)(19). Reaction at this center is assumed to take place with unit probability upon first encouilter-i.e., the reaction is assumed to be strictly diffusion controlled.…”

“…A general procedure for dealing with stochastic processes on finite d-dimensional lattices was presented in 1958 by Montroll and Shuler (11). This procedure, coupled with methods derived from the theory offinite Markov processes, was implemented in a recent series of papers to study a variety of theoretical and physical problems on finite lattice systems (12)(13)(14)(15)(16)(17)(18)(19). In the present study, we shall assign one site of each regular polyhedron to be a "reaction center" and then calculate the site-specific and overall mean walk length (a measure of the "reaction time"; see later text) of a coreactant diffusing on the surface to that reaction center.…”