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

TL;DR

This paper develops a formalism to analyze the statistical properties of functionals of a particle's path in a one-dimensional random potential, revealing how disorder significantly alters these properties.

Contribution

The paper introduces a new formalism for calculating the statistical distributions of functionals of diffusing particles in quenched random potentials, with explicit examples.

Findings

Disorder drastically modifies the distributions of local and occupation times.

Explicit distributions for local time, inverse local time, occupation time, and inverse occupation time are derived.

Disorder effects are significant in many cases.

Abstract

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.