Periodical Systems

In this chapter, we studied the diffusion of Brownian particles through long, narrow, periodic channels. To such end, we simplify the effective diffusivity by means of the so-called Lifson-Jackson formula, which considerably simplifies the problem. This formula is an exact result for the one-dimensional Smoluchowski equation in the presence of a periodic potential and periodic diffusivity. Consequently, the effective diffusion coefficient becomes constant once the geometrical parameters of the channel, then, the description of the system can be done with a Fick-Jacobs equation, instead of a Fick-Jacobs-Zwanzig equation. We also present the application of the formula for explicit periodic channels in two and three dimensions, as well as a system formed by cylindrical obstacles.