Intermittent free diffusion in the presence of sparse obstacles: mean obstacle encounter time

Particle diffusion in the presence of sparse obstacles may be considered as a sequence of relatively long intervals, during which the particle diffuses in regions free of obstacles, separated by relatively short intervals during which the particle suffers multiple collisions with an obstacle. The present paper focuses on the mean duration of the short intervals, called mean encounter time, assuming that the obstacles are identical spheres. Based on scaling arguments, one can deduce that this time is proportional to the ratio of the square of the obstacle radius and the particle diffusivity. We derive an expression for the mean encounter time, which shows that the proportionality coefficient is 1/6.