Nonlinear Brownian motions in gases: Kinetic equations and relaxation processes
Phys. Rev. E 114, 044139 – Published 27 October, 2026
DOI: https://doi.org/10.1103/qm4r-yqnc
Abstract
The behavior of particles subject to a nonlinear drag force in a medium (nonlinear Brownian motion) is usually treated by means of the Fokker-Planck equation proposed by Klimontovich [Phys. Usp. 37, 737 (1994)]. However, the derivation of such an equation appears to be questionable since it involves unproved assumptions and does not take into account any relation between the assumptions and approximations leading (in kinetic theory) to a velocity-dependent friction coefficient and those necessary to obtain an appropriate kinetic equation. In spite of this, as far as we know, the limits of validity of the said equation have not yet been investigated. Such circumstances pose the general problem: what is (if any) the appropriate (approximate) differential kinetic equation which is really compatible with the Langevin equation for nonlinear Brownian motions? To adequately solve this problem, previous papers dealing with the behavior of heavy particles in light gases are reexamined. The study leads to the final conclusions that (1) the heavy-particle speed dependence of the friction coefficient (and of the fluctuating force), in the Langevin equation, makes it very difficult to derive an appropriate Fokker-Planck equation for nonlinear Brownian motion, particularly in the most general three-dimensional case, and (2) the searched appropriate (approximate) differential kinetic equation for the heavy particles really exists and is the fourth-order equation following from the Boltzmann equation for a quasi-Rayleigh gas. Then it is shown that, on the contrary, the equation for the heavy-particle mean velocity relaxation following from the Fokker-Planck equation proposed by Klimontovich conflicts with that directly derived from the Langevin equation, a circumstance which now proves the inadequacy of the Klimontovich approach. Subsequently, some relaxation processes of particular interest are studied. So both the correct time-dependent mean velocity and the mean distance traveled by an anisotropic swarm of heavy particles are obtained, and the results are discussed assuming an interaction force between heavy- and gas-particles proportional to inverse powers of their mutual distance. Moreover, the correct relaxation equation for the heavy-particle mean-square speed is also achieved, and the difference with the corresponding result following from the Klimontovich theory is pointed out. An approximate method of solution of the said correct relaxation equation is also proposed. Finally, the limits of validity of the present theory are discussed.
