Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential

Sanjib Sabhapandit 1, 2, Satya N. Majumdar 1, Alain Comtet 1, 2

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 73 (2006) 051102

We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size $t$. We compute the disorder average distributions of the local time, the inverse local time, the occupation time and the inverse occupation time, and show that in many cases disorder modifies the behavior drastically.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Unite mixte de service de l'institut Henri Poincaré (UMSIHP),
    CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie