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

J. T. Chalker·Z. Jane Wang
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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, $ \sim t^{2/d}$ in dimension d>2, associated with the imaginary parts of eigenvalues.

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J. T. Chalker, Z. Jane Wang. 1999-06-18. Spectrum of the Fokker-Planck operator representing diffusion in a random velocity field. https://doi.org/10.1103/physreve.61.196

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