Occupation times for single-file diffusion

Olivier Bénichou 1 Jean Desbois 2

Journal of Statistical Mechanics: Theory and Experiment, IOP Science, 2015, pp.P03001

We consider a file of identical Brownian particles moving on the same axis x'Ox without crossing each other. They all start from the origin O at time t = 0 and are stopped at some time t. Denoting by T the time spent on the half-line [Ox) by a given particle of the line, we establish analytical formulae for the first two moments 〈T〉 and 〈T2〉. In particular, considering the limit of an infinite number of particles, we get, for the 'middle' particle (J0 is a Bessel function). This result (and also numerical simulations) shows that the distribution of T, though being close to it, is not fully a constant one.

  • 1. LPTMC - Laboratoire de Physique Théorique de la Matière Condensée
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques