Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

Abstract

This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.

A 3D vortex formed by trajectories of the Navier–Stokes equation

“…Nay, the Lie group is a useful tool for physicists to deal with stochastic processes (note the detailed comments in [20,21]). The classic examples of such works used the basic idea of the Lie group to study the positional and orientational distributions for some random walks [22,23], to reveal the conformational statistics of semi-flexible molecular chains [24] and polymers [25], and to characterize the bio-molecular such as conformation of DNA, helix-helix interactions in proteins, and so on [26,27]. Equally illuminating works are those which obtained the description of the drift-diusion motions of the electrons in semiconductors with the aid of the same reduction method [28], the characterization of collision dynamics via transforming the equations of motion into the purely algebraic ones [29], and discussions on the inhomogeneous diusions by simplifying the Fokker-Planck equations with the so-called potential symmetries that are derived from computing Lie groups of point transformations [30,31].…”