Winding statistics of a Brownian particle on a ring

Anupam Kundu 1 Alain Comtet 1, 2 Satya N. Majumdar 1

Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2014, 47 (38), pp.385001

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.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. IHP - Institut Henri Poincaré