Semiclassical theory of quantum diffusion in 1D: A stochastic process for the Landauer equation

Abstract

With the help of the quantum theory of scattering, a set of simultaneous difference equations is proposed, defining a diffusive stochastic process (correlated walk), which describes a semiclassical, one-dimensional, tunneling diffusion process in a periodic lattice. The jump probabilities are just the quantum transmission coefficients of the unit cell. With this process, we prove that by quantum tunneling or scattering above the potential, the particles diffuse in a one-dimensional lattice with a diffusion coefficient given by the Landauer formula. Next, the theory is generalized to include higher dimensions, and a square lattice case is shown as an example. There we show that the Landauer formula will not be obtained for 2D or 3D cases. Finally, examples are given, in which the Landauer formula is also valid for other cases of one-dimensional Markovian correlated walks.