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

## TL;DR

This paper evaluates a path integral for Brownian loops in 3D space to quantify the correlation between their linking numbers with two arbitrary curves, revealing how these correlations vary with the curves' proximity.

## Contribution

It provides a novel calculation of the kinetic path integral for linking numbers of Brownian loops with two curves, extending previous 2D results to 3D.

## Key findings

- Correlation varies from zero to infinity depending on curve proximity
- Path integral resembles mutual inductance in magnetostatics
- Results depend on the duration of the Brownian path

## 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].

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Source: https://tomesphere.com/paper/1905.03858