From normal to anomalous diffusion in comb-like structures in three dimensions.

Diffusion in a comb-like structure, formed by a main cylindrical tube with identical periodic dead ends of cylindrical shape, occurs slower than that in the same system without dead ends. The reason is that the particle, entering a dead end, interrupts its propagation along the tube axis. The slowdown becomes stronger and stronger as the dead end length increases, since the particle spends more and more time in the dead ends. In the limiting case of infinitely long dead ends, diffusion becomes anomalous with the exponent equal to 1/2. We develop a formalism which allows us to study the mean square displacement of the particle along the tube axis in such systems. The formalism is applicable for an arbitrary dead end length, including the case of anomalous diffusion in a tube with infinitely long dead ends. In particular, we demonstrate how intermediate anomalous diffusion arises when the dead ends are long enough.