Diffusion in classical periodic systems: The Smoluchowski equation approach

The classical diffusion of a particle in a periodic system is studied employing the Smoluchowski equation with an external periodic field of force. Assuming a two-dimensional rectangular symmetry and an external potential of a simple cosine shape, the resulting eigenvalue problem leads to a Hill equation, which is solved numerically. The dynamic structure factor and its full width at half maximum (fwhm) are calculated up to large values of the momentum transfer extending in several Brillouin zones. It is shown that, decreasing the amplitude of the potential, the Smoluchowski equation describes a diffusive process which changes continuously from a jump to a liquid-like regime, with an intermediate behaviour which has some characteristics of both. In this context, recent scattering experiments on systems (premelting surfaces and absorbed monolayers) which seem to show such a behaviour, are briefly discussed.