The correlated linking numbers of a Brownian loop with two arbitrary curves

Abstract

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to (plus or minus) infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying curves in magnetostatics, a double integral, but dependent on a single extra parameter, the duration t of the path. The result for the equivalent two-dimensional problem was given previously [Hannay 2018].