Winding statistics of a Brownian particle on a ring

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.

15 pages, 7figs

References in corpus (6)

• Universal Record Statistics of Random Walks and Lévy Flights
• Record statistics and persistence for a random walk with a drift
• Functionals of the Brownian motion, localization and metric graphs
• Universal statistics of longest lasting records of random walks and Lévy flights
• Quantum oscillations in mesoscopic rings and anomalous diffusion
• SLE on doubly-connected domains and the winding of loop-erased random walks

Cited by in corpus (2)

• Transport properties of diffusive particles conditioned to survive in trapping environments
• Winding number of a Brownian particle on a ring under stochastic resetting