The theory of time-nonlocal random processes formulated in terms of the non-Markovian Fokker–Planck equation is used to describe the results of numerical simulations of particle diffusion in the random longitudinal field with given statistical properties. The simulations of particle motion were performed for the wide range of particle velocity and random field parameters. It is confirmed that conventional quasilinear theory in the approximation disregarding the time and velocity dependence of the diffusion coefficient in the velocity space can be used only in the case of small intensity and large width of turbulent field spectrum. The increase of the intensity as well as the decrease of the spectral width lead to considerable deviation of the results of simulations (such as saturation and frequent oscillation of the mean-square velocity displacement) from the predictions of the quasilinear theory. It is shown that in the case of small intensities these deviations can be successfully described in terms of non-Markovian generalization of the quasilinear approximation. In the case of high field intensity the description of these features would require more consistent account for the diffusion coefficient velocity dependence and time-nonlocal effects.

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Diffusion in velocity space of test particles undergoing external random electric fields with spectra varying from low intensive and broad to high intensive and narrow (peaked) is considered. It is shown that to achieve consistency between simulation and prediction of the microscopic model, which is reduced to Fokker–Planck-type equation, it is necessary, in the case of peaked spectrum, to account for temporal variation of diffusion coefficient occurring in the early stage. An analytical approximation for the solution of the Fokker–Planck equation with a time and velocity dependent diffusion coefficients is proposed.

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Velocity distribution of dust particles localized in a plasma sheath near an electrode was found in a number of experiments. Velocity dispersion indicated that the kinetic temperature of dust grains significantly exceeds the temperature of plasma environment. Consequently, the question arose about the stochastic mechanisms of anomalous heating of grains. We propose the model in which the kinetic energy is due to the significant potential energy that grains have at the moment of their release from the crystalline structure on melting. Stochastic processes only modify the regular motion of dust grains, forming a velocity distribution similar to а Gaussian.