Eigenfunction approach to the persistent random walk in two dimensions

Abstract

The Fourier-Bessel expansion of a function on a circular disc yields a simple series representation for the end-to-end probability distribution function w(R,phi) encountered in a planar persistent random walk, where the direction taken in a step depends on the relative orientation towards the preceding step. For all but the shortest walks, the proposed method provides a rapidly converging, numerically stable algorithm that is particularly useful for the precise study of intermediate-size chains that have not yet approached the diffusion limit. As a practical application, we examine the force-extension diagram of various planar polymer chains. With increasing joint stiffness, a marked transition from rubber-like behaviour to a form of elastic response resembling that of a flexible rod is observed.