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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 γ 0 one includes a function γ(v)∼v 2α . 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 F fric=−γ(v)v∼v 3. 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.

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