Generalized Fokker-Planck Kinetic Model

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Abstract
We study some consequences of a nonlinear kinetic model that has the structure of a constant-collision-time, Fokker-Planck equation. The collision operator, however, satisfies momentum and energy conservation in addition to particle conservation. One can find the relaxation spectrum for the nonlinear as well as linear problem. It consists of an infinite sequence of decays with the spectrum of the quantum harmonic oscillator. It is also possible to solve exactly the initial-value problem for the linearized kinetic equation. The Green's function for the complete operator obeys a degenerate integral equation containing the Green's function of the usual Fokker-Planck equation. We find the exact time-dependent density correlation function and thus the inelastic scattering cross section that determines the neutron scattering and the Brillouin scattering of light from the system.