Velocity and displacement statistics in a stochastic model of nonlinear friction showing bounded particle speed

TL;DR

This paper introduces a nonlinear friction diffusion model with a maximum particle speed, analyzing velocity and displacement statistics, including effects of a constant drift force, with some analytical solutions and numerical methods.

Contribution

It presents a novel nonlinear friction model with a maximum particle speed and derives analytical results for stationary velocity distributions and drift effects.

Findings

Stationary velocity distributions can be derived analytically.

Displacement statistics generally require numerical methods.

Analytical progress is made for a special maximum particle speed case.

Abstract

Diffusion of colloidal particles in a complex environment such as polymer networks or biological cells is a topic of high complexity with significant biological and medical relevance. In such situations, the interaction between the surroundings and the particle motion has to be taken into account. We analyze a simplified diffusion model that includes some aspects of a complex environment in the framework of a nonlinear friction process: at low particle speeds, friction grows linearly with the particle velocity as for regular viscous friction; it grows more than linearly at higher particle velocities; finally, at a maximum of the possible particle speed the friction diverges. In addition to bare diffusion, we study the influence of a constant drift force acting on the diffusing particle. While the corresponding stationary velocity distributions can be derived analytically, the displacement statistics generally must be determined numerically. However, as a benefit of our model, analytical progress can be made in one case of a special maximum particle speed. The effect of a drift force in this case is analytically determined by perturbation theory. It will be interesting in the future to compare our results to real experimental systems. One realization could be magnetic colloidal particles diffusing through a shear-thickening environment such as starch suspensions, possibly exposed to an external magnetic field gradient.