Velocity-dependent friction and diffusion for grains in neutral gases, dusty plasmas and active systems

A self-consistent and universal description of friction and diffusion for Brownian particles (grains) in different systems, such as gases with Boltzmann collisions, dusty plasmas with ion absorption by grains, and for active particles (e.g., cells in biological systems), is suggested on the basis of the appropriate Fokker–Planck equation. Restrictions for application of the Fokker–Planck equation to the problem of velocity-dependent friction and diffusion coefficients are found. A general description for these coefficients is formulated on the basis of a master equation. Relation between the diffusion coefficients in the coordinate and velocity spaces is found for active (capable to transfer momentum to the ambient medium) and passive particles in the framework of the Fokker–Planck equation. The problem of anomalous space diffusion is formulated on the basis of the appropriate probability transition (PT) function. The method of fractional differentiation is avoided to construct the correct probability distributions for arbitrary distances, which is important for applications to different stochastic problems. A general equation for the time-dependent PT function is formulated and discussed. The generalized friction in velocity space is determined and applied to describe the friction force itself as well as the drag force in the case of a non-zero driven ion velocity in plasmas. Negative friction (due to ion scattering) on grains exists and can be realized for the appropriate experimental conditions.