Kinetic equation approach to the description of quantum surface diffusion: non-Markovian effects versus jump dynamics

We consider surface diffusion of a single particle, which performs site-to-site underbarrier hopping, fulfils intrasite motion between the ground and the first-excited states within a quantum well, and interacts with surface phonons. We obtain a chain of quantum-kinetic equations for one-particle distribution functions and nonequilibrium hopping probabilities. The generalized diffusion coefficients are derived, and the generic non-Markovian diffusion equation is written down both for the infinite lattice model and in the continuous media limit. In the latter case, the one-particle distribution function obeys the telegrapher's equation, which could give us a nonmonotonic behavior of the intermediate distribution functions at large spatial gradients. In a weak-coupling limit, if the energy of level splitting is comparable with the temperature, there are also pronounced oscillations of the generalized diffusion coefficients. The recrossing/multiple crossing phenomena, a problem of long tails of the generalized diffusion coefficients, as well as a mapping into the next- to the nearest-neighbors hopping regime, are discussed.