Theory of diffusion in finite random media with a dynamic boundary condition

In the presence of a time-periodic incoming flow the diffusion problem on finite random media has been studied. Particular importance has been stressed on different boundary conditions (reflecting or absorbing). The problem has been worked out by generalizing the finite effective-medium approximation (FEMA). Thus a perturbative theory, in the time-asymptotic regime, has been built up in the Laplace representation (small-u parameter) for weak and strong site disorder. This theory separates in a natural way the contribution given by the effective medium (from other higher-order corrections) which appears as the zeroth-order step in the perturbation scheme. Asymptotic results for the current of probability (inside the finite domain) are obtained for different time-dependent incoming external flows. Exact results beyond FEMA are obtained for the low-frequency behavior of the evolution equation for the averaged Green's function on a finite lattice. Monte Carlo simulations have been carried out in order to compare them with our theoretical predictions. In the absence of an incoming external flow, the problem of the first passage time distribution through only one frontier (in a random media) and for different boundary conditions has been revisited.