Diffusive processes in systems with geometrical constraints: from lattice models to continuous channels

We focus on the Brownian motion within channels with varying cross – section as well as on the diffusion on branched structures, a subject particularly relevant for the modeling of the passive transport process within living cells, which are characterized by a naturally constrained environment and the motion, at least on the length scales of the macromolecules, is well described by diffusion rather than bulk flow. In particular, we studied the occurrence of anomalous transport in a variety of branched structures and fractal trees. We proved that the Fluctuation Dissipation Relation (FDR) can be extended, in some cases, to the anomalous transport regime, at least within the linear response approximation. On the other hand, We pointed out how the FDR can be broken by choosing properly the branching of the analyzed structure, as a consequence of the emergence of an “entropic” drift due to the high ramifications introduced. This result suggests that FDRs are more sensitive to the geometrical structure rather than to the details of the dynamics. We also analyzed, in contrast to those examples showing anomalous diffusion, a series of situations characterized by a standard scaling of the MSD (and/or of the higher order moments), however with a non Gaussian probability density, thus showing how standard diffusion is not always Gaussian.