Mean square winding angle of Brownian motion around an impenetrable cylinder

Hannay, J H and Wilkinson, Michael (2022). Mean square winding angle of Brownian motion around an impenetrable cylinder. Journal of Physics A: Mathematical and Theoretical, 55(23), article no. 234007.

DOI: https://doi.org/10.1088/1751-8121/ac6af7

Abstract

An exact formula is derived, as an integral, for the mean square value of the winding angle ϕ (with −∞< ϕ <∞) of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius <i>a</i>, having started at radius R(><i>a</i>) from the axis. Strikingly, for the simpler problem with <i>a</i>= 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 straightaway 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 <i>a</i><sup>2</sup> ≪ <i>R</i><sup>2<sup>, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.

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