Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer-Edwards-Prager-Frisch model revisited

TL;DR

This paper revisits classical models of winding angle distributions in planar Brownian motion and ring polymers, providing new exact calculations for linking numbers and generalizations involving obstacles and confined spaces.

Contribution

It offers a unified Green function approach to re-derive key results and introduces new exact calculations for linking number distributions and obstacle geometries.

Findings

Exact expectation value of the surface area with linking number n

Re-derivation of classical winding angle distributions

Generalizations to finite diameter obstacles and cavities

Abstract

Using a general Green function formulation, we re-derive, both, (i) Spitzer and his followers results for the winding angle distribution of the planar Brownian motion, and (ii) Edwards-Prager-Frisch results on the statistical mechanics of a ring polymer entangled with a straight bar. In the statistical mechanics part, we consider both cases of quenched and annealed topology. Among new results, we compute exactly the (expectation value of) the surface area of the locus of points such that each of them has linking number $n$ with a given closed random walk trajectory ($=$ ring polymer). We also consider the generalizations of the problem for the finite diameter (disc-like) obstacle and winding within a cavity.