Random walk of multiple agents in a confined space

Abstract

While most literature on random walk involves unconfined motion of a single random walker, self-organization of real systems is normally driven by space-confined motion of a collection of agents. Here, we study the effect of confinement and density of N randomly moving agents in the time evolution of the mean squared distance R (T) = < r² >, where r is the displacement from the initial location. The motion of agents in a square lattice of dimension L x L follows Von-Neumann type of neighborhood. We establish the existence of a universal saturation point equal to Rsat = R(T → ∞) ~ 0.33 L² , and provide an analytic approximation for this limit. The saturation is shown to be independent of the lattice size and the density ρ = N/L² of agents that are moving simultaneously. The approach to the equilibrium however is shown to evolve accurately following R(T) = Rsat(1 - e⁻ᵇᵗ), where b is proportional to L² (1 - ρ).