Mean square winding angle of Brownian motion around an impenetrable cylinder

TL;DR

This paper derives an exact integral formula for the mean square winding angle of 3D Brownian motion around an impenetrable cylinder, addressing divergence issues in the zero-radius case and analyzing limits for narrow cylinders.

Contribution

It provides the first exact integral expression for the mean square winding angle around a finite-radius cylinder, extending previous approximate methods and resolving divergence problems.

Findings

Exact integral formula for mean square winding angle derived.

Analysis of limits for narrow cylinders (a << R).

Addresses divergence in zero-radius case by considering finite thickness.

Abstract

An exact formula is derived, as an integral, for the mean square winding angle of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius a, having started at radius R(>a) from the axis. Strikingly, for the simpler problem with a=0, the mean square winding angle around a straight line, is long known to be instantly infinite however far away the starting point lies. the fractally small, fast, random walk steps of mathematical Brownian motion allow unbounded windings around the zero thickness of the straight line. A remedy if it is required, is to accord the line non-zero thickness, an impenetrable cylinder, as analysed here. The problem straight away reduces to a 2D one of winding around a disc in a plane since the axial component of the 3D Brownian motion is independent of the others. After deriving the exact mean square winding angle, the integral is evaluated in the limit of a narrow cylinder a<<R, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.