Canonical ensemble of particles in a self-avoiding random walk

Abstract

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ℝ^d. The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r₀, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ℝ^d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r₀. We describe a method for computing the asymptotic form of the probability density W_n(r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W_n(r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ℝ^d for 2 ≤ d < 4.