  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.
