A quantum random‐walk model for tunneling diffusion in a 1D lattice. A quantum correction to Fick’s law

With the help of quantum‐scattering‐theory methods and the approximation of stationary phase, a one‐dimensional coherent random‐walk model which describes both tunneling and scattering above the potential diffusion of particles in a periodic one‐dimensional lattice is proposed. The walk describes for each lattice cell, the time evolution of modulating amplitudes of two opposite‐moving Gaussian wave packets as they are scattered by the potential barriers. Since we have a coherent process, interference contributions in the probabilities bring about strong departures from classical results. In the near‐equilibrium limit, Fick’s law and its associated Landauer diffusion coefficient are obtained as the incoherent contribution to the quantum current density along with a novel coherent contribution which depends on the lattice properties as [(1−R)/R]1/2.