A simple example of anomalous diffusion

The motion of a particle confined within a strip (“flat pipe”) with diffusely-reflecting walls is analyzed in detail and provides a simple example of the interplay between randomness and geometrical constraint. The mean square dispersion Δx2 of the position is exactly found in various cases, allowing for a detailed asymptotic analysis. For elastic bounces, Δx2 is shown to behave as t2/ln t at large times, with logarithmic corrections. This anomalous behaviour is rather robust in the sense that it still occurs for thermal bounces. On the other hand, when the particle has a high probability to emerge from a bounce with a very small velocity, Δx2 follows various anomalous time behaviours which, nevertheless, are always superdiffusive.