Eigenfunction approach to the persistent random walk in two dimensions

TL;DR

This paper introduces a Fourier-Bessel expansion method for analyzing the end-to-end distribution of two-dimensional persistent random walks, providing a stable and efficient way to study intermediate chain lengths and their mechanical properties.

Contribution

It presents a novel Fourier-Bessel expansion approach for the persistent random walk in two dimensions, enabling accurate analysis of intermediate chain sizes and their force-extension behavior.

Findings

Method yields rapidly converging, numerically stable results.

Transition from rubber-like to elastic response with increased stiffness.

Applicable to studying polymer chain mechanics.

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.