Fundamental Function for the Lattice-based Random Walk Simulation in One Dimension

Abstract

diffusion-influenced reactions attract increasing attention. It is well-known that diffusion-influenced reac-tions can be described by lattice-based random walk Monte Carlo simulations.1-6 Since the continuum limit of the random walk model is closely related to the diffusion pro-cess, we can straightforwardly simulate general diffusion-reaction systems with the lattice-based Monte Carlo simulation. Although the theories of the random walk on the lattice points have been regularly reported,1,7 the analytical results for diffusion-reaction systems on lattice points have been relatively rare. In this Note, we find the fundamental distribution function of a random walker on the one-dimensional lattice with a reactive trap. This distribution function leads to a discrete version of the survival probability, which can be reduced to the well-known survival probability with the absorbing or Smoluchowski boundary condition.8 Non-reactive Random Walk Model in One Dimension Firstly, we consider the random walk model without a reactive trap. Let PN(x, n; x0) be the non-reactive probability that the walker is observed at x after n steps with the starting position x0 on the one-dimensional lattice. Suppose that jumps to the left and to the right occur with probability p and q = 1−p, respectively. After the first step, we have PN(x0− 1,1; x0) = p and PN(x0+1,1; x0) = q, and after two steps, PN(x0−2,2; x0) = p2, PN(x0,2; x0) = 2pq, and PN(x0+2,2; x0) = q2, and so on. The probability that the walker moved k times to the left and n – k times to the right is given by the binomial distribution PN(x, n; x0) = n Ckp kqn−k (1) By definition, k = (x0 + n − x)/2 should be an integer satisfying or, and is not zero only when k = n, n−1,..., 0 or x = −n + x0, −n + x0 + 2,..., n + x0. If the jump prob-abilities p and q are the same as p = q = 1/2, the probability after n steps is given b