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

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, compare our results with numerical simulations, and relate them to the time evolution of particle density. For strong disorder and short times, we find a novel time dependence of the mean-square displacement: $〈r^2〉\sim t^{2/d}$ in dimension $d>\mathrm{}2\mathrm{}$.