Winding angle distribution of 2D random walks with traps

TL;DR

This paper analytically investigates the asymptotic distribution of winding angles for 2D Brownian particles in trap-laden environments, revealing a specific exponential decay and a universal scaling form.

Contribution

It provides a new analytical description of the winding angle distribution for 2D Brownian motion with traps, including asymptotic behavior and scaling laws.

Findings

Probability decays as exp(-c√t) for large t

Distribution scales with n/√t as (1+x^2)^{-1}

Long-time tail corresponds to particles avoiding traps

Abstract

We study analytically the asymptotic behaviour of the average probability P(n,t) for the trajectory of a 2D Brownian particle wandering in the presence of randomly distributed traps to wind n times around a given point after a time t. It is shown that P(n,t)\sim\exp(-c\sqrt{t}) (1+x^2)^{-1} with x\sim n/\sqrt{t}, where the first exponent represents a well known long-time tail of the probability that a particle will not be trapped.