Diffusion in a Random Velocity Field: Spectral Properties of a Non-Hermitian Fokker-Planck Operator

TL;DR

This paper investigates the spectral characteristics of a non-Hermitian Fokker-Planck operator modeling particle diffusion in a quenched random velocity field, revealing eigenvalue distributions and their implications for particle dynamics.

Contribution

It provides analytical calculations of eigenvalue density and Green's functions for weak disorder in dimensions greater than two, linking spectral properties to diffusion behavior.

Findings

Eigenvalues occupy a finite area in the complex plane.

Derived eigenvalue density and Green's function for weak disorder.

Results agree with numerical simulations.

Abstract

We study spectral properties of the Fokker-Planck operator that describes particles diffusing in a quenched random velocity field. This random operator is non-Hermitian and has eigenvalues occupying a finite area in the complex plane. We calculate the eigenvalue density and averaged one-particle Green's function, for weak disorder and dimension d>2. We relate our results to the time-evolution of particle density, and compare them with numerical simulations.