Effect of a forbidden site on a d-dimensional lattice random walk

摘要

We study the effect of a single excluded site on the diffusion of a particle undergoing a random walk in a d-dimensional lattice. The determination of the characteristic function allows us to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) ((x) over right arrow) (t) and the mean square deviation ((x) over right arrow (2))(t) - ((x) over right arrow)(2)(t). In contrast to the one-dimensional case, where ((x) over right arrow)(t) diverges at infinite times ((x) over right arrow)(t) similar to t(1/2)) and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d greater than or equal to 2, ((x) over right arrow)(t) is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d = 2) still occur. Finally, the continuum space version of the model is analysed; it is shown that d = 1 is the lower dimensionality above which all the effects of the forbidden site are irrelevant. [References: 10]