The diffusion coefficient of nonlinear Brownian motion

Abstract

Nonlinear Brownian motion (BM) refers to cases where the damping constant and possibly also the noise intensity in the Langevin equation depend on the velocity of the particle. Such velocity dependence is encountered in cases where Stokes' linear friction law does not apply, for relativistic Brownian particles, and for models of active motion of biological objects. For an arbitrary velocity dependence of damping and noise intensity, the diffusion coefficient can be given in terms of quadratures. We evaluate and discuss this quadrature formula for the three different cases. For a nonlinear friction (being larger at high speed than expected from Stokes' friction) and for the relativistic BM we obtain in general diffusion coefficients that are smaller than those for linear BM. The diminished diffusion in these equilibrium systems has different physical reasons. For the nonequilibrium model of active motion we demonstrate that diffusion can be minimized at a finite noise intensity. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.