The Local and the Occupation Time of a Particle Diffusing in a Random Medium

Satya N. Majumdar 1, Alain Comtet 2, 3

Physical Review Letters 89 (2002) 060601

We consider a particle moving in a one dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean value) and the occupation time (spent above its mean value) within an observation time window of size t. The random part of the potential is same as in the Sinai model, i.e., the potential itself is a random walk in space. In the absence of the random potential, these distributions have three typical asymptotic behaviors depending on whether the deterministic potential is unstable, stable or flat. These asymptotic behaviors are shown to get drastically modified when the random part of the potential is switched on leading to the loss of self-averaging and wide sample to sample fluctuations.

  • 1. Laboratoire de Physique Quantique (LPQ),
    CNRS : UMR5626 – Université Paul Sabatier - Toulouse III
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. IHP,
    Institut Henri Poincaré