Spectrum of the Fokker-Planck operator representing diffusion in a random velocity field

TL;DR

This paper analyzes the spectral properties of a Fokker-Planck operator modeling particles in a random velocity field, revealing a non-trivial eigenvalue distribution and a novel short-time diffusive behavior in higher dimensions.

Contribution

It provides an analytical calculation of the eigenvalue density and Green function for the Fokker-Planck operator using diagrammatic and mean-field methods, especially in the weak-disorder regime.

Findings

Eigenvalue density forms a wedge in the complex plane enclosing the negative real axis.

Particle motion is diffusive at long times.

Short-time behavior shows a mean-square displacement scaling as t^{2/d} for d>2.

Abstract

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. {\bf 79}, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well-controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is non-zero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time-dependence of the mean-square displacement, $<r^2> \sim t^{2/d}$ in dimension d>2, associated with the imaginary parts of eigenvalues.