Diffusion in the presence of a pole: From the continuous Gaussian to a discrete lattice model

Abstract

The continuous Gaussian model with delta function potentials which has been used to describe the effects that interacting centers (poles) bring to the diffusion of a random walker, provides an exact solution only in dimension d = 1 . Its application in dimension d ⩾ 2 is in need of introduction of cutoffs coming from the finite sizes of poles. We therefore formulate an analogous discrete model in the d -dimensional lattice space where no cutoffs or their regularization are necessary. The discrete model is amenable to numerical treatment and the Green's function, its first moments and properties derived from them can be determined numerically to all dimensionalities. In the one-dimensional (1-D) case, in the presence of a single center, the outcomes of the Gaussian model are compared, with the corresponding results of the discrete lattice model. The possibility of extension of the lattice model to higher dimension permits the study of the two-dimensional (2-D) case too. The intensities of interactions considered before to obtain positive values are extended to negative values in order to describe attractions also.