Bulk Mediated Surface Diffusion: Finite System Case

We address the dynamics of adsorbed molecules (a fundamental issue in surface physics) within the framework of a Master Equation scheme, and study the diffusion of particles in a finite cubic lattice whose boundaries are at the $z=1$ and the $z=L$ planes where $L = 2,3,4,...$, while the $x$ and $y$ directions are unbounded. As we are interested in the effective diffusion process at the interface $z = 1$, we calculate analytically the conditional probability for finding the system on the $z=1$ plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement.