Diffusion Coefficient of a Brownian Particle with a Friction Function Given by a Power Law

Nonequilibrium biological systems like moving cells or bacteria have been phenomenologically described by Langevin equations of Brownian motion in which the friction function depends on the particle’s velocity in a nonlinear way. An important subclass of such friction functions is given by power laws, i.e., instead of the Stokes friction constant γ0 one includes a function γ(v)∼v2α. Here I show using a recent analytical result as well as a dimension analysis that the diffusion coefficient is proportional to a simple power of the noise intensity D like D(1−α)/(1+α) (independent of spatial dimension). In particular the diffusion coefficient does not depend on the noise intensity at all, if α=1, i.e., for a cubic friction Ffric=−γ(v)v∼v3. The exact prefactor is given in the one-dimensional case and a fit formula is proposed for the multi-dimensional problem. All results are confirmed by stochastic simulations of the system for α=1, 2, and 3 and spatial dimension d=1, 2, and 3. Conclusions are drawn about the strong noise behavior of certain models of self-propelled motion in biology.