Diffusion Through a Network of Compartments Separated by Partially-Transmitting Boundaries

We study the random walk of a particle in a compartmentalized environment, as realized in biological samples or solid state compounds. Each compartment is characterized by its length L and the boundaries transmittance T. We identify two relevant spatio-temporal scales that provide alternative descriptions of the dynamics: (i) the microscale, in which the particle position is monitored at constant time intervals; and (ii) the mesoscale, in which it is monitored only when the particle crosses a boundary between compartments. Both descriptions provide—by construction—the same long time behavior. The analytical description obtained at the proposed mesoscale allows for a complete characterization of the complex movement at the microscale, thus representing a fruitful approach for this kind of systems. We show that the presence of disorder in the transmittance is a necessary condition to induce anomalous diffusion, whereas the spatial heterogeneity reduces the degree of subdiffusion and, in some cases, can even compensate for the disorder induced by the stochastic transmittance.