Winding statistics of a Brownian particle on a ring

TL;DR

This paper derives exact analytical distributions for the total and net turns of a Brownian particle on a ring, revealing Gaussian scaling and large deviation behaviors, supported by numerical simulations.

Contribution

It introduces a renewal-based method to analytically compute winding distributions, offering an alternative to path integral techniques.

Findings

Distributions have Gaussian scaling forms for large t

Large deviation functions characterize rare fluctuations

Numerical simulations confirm analytical results

Abstract

We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal properties of Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we show that these distributions have Gaussian scaling forms. We also compute large deviation functions associated to these distributions characterizing atypically large fluctuations. We provide numerical simulations in support of our analytical results.