One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

TL;DR

This paper investigates one-dimensional classical diffusion in a Gaussian random force field with weak absorbers, revealing a power-law decay of return probability and discussing localization properties, contrasting with standard exponential decay.

Contribution

It introduces a novel analysis linking the Fokker-Planck operator to a supersymmetry-breaking quantum Hamiltonian, deriving a power-law decay in the presence of a random force field and absorbers.

Findings

Return probability decays as a power law $t^{- oot{2 ho/g}}$

Contrasts with exponential decay in absence of force field

Discusses localization properties of the quantum Hamiltonian

Abstract

A one-dimensional model of classical diffusion in a random force field with a weak concentration $\rho$ of absorbers is studied. The force field is taken as a Gaussian white noise with $\mean{\phi(x)}=0$ and $\mean{\phi(x)\phi(x')}=g \delta(x-x')$. Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law $\smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}}$ (to be compared with the usual Lifshits exponential decay $\exp{-(\rho^2t)^{1/3}}$ in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.