Anomalous diffusion in velocity space

The phenomenon of anomalous diffusion in the momentum space is considered on the basis of the appropriate probability transition function (PTF). New general equation for description of the kinetics of particles is derived on basis of the approach similar to one in the coordinate space (Trigger et al., 2005) [1]. This equation is a generalization of the Fokker–Planck equation for the cases of the specific PTF with the long tails in the momentum space. The results are applied to the particular case of the Boltzmann-type PTF kernel with arbitrary differential cross-section and the prescribed distribution function of the another type of colliding particles. The simplest case of diffusion of hard spheres in the gas of light particles is described. The obtained results permit to analyze various situations when the probability transition function has a long tail in the momentum space. The effective friction and diffusion coefficients are found.