Size-independence of statistics for boundary collisions of random walks and its implications for spin-polarized gases

TL;DR

This paper demonstrates that the statistical distribution of boundary collisions in a bounded random walk is size-independent in the large time limit, with significant implications for spin-polarized gases and depolarization effects.

Contribution

It provides the full statistical distribution of boundary collisions for 1D random walks and shows size-independence of fluctuations, impacting understanding of depolarization in spin gases.

Findings

Collision fluctuations are size-independent at large times.

Boundary effects do not diminish with system size.

Implications for magnetic impurity-induced depolarization in gases.

Abstract

A bounded random walk exhibits strong correlations between collisions with a boundary. For an one-dimensional walk, we obtain the full statistical distribution of the number of such collisions in a time t. In the large t limit, the fluctuations in the number of collisions are found to be size-independent (independent of the distance between boundaries). This occurs for any inter-boundary distance, including less and greater than the mean-free-path, and means that this boundary effect does not decay with increasing system-size. As an application, we consider spin-polarized gases, such as 3-Helium, in the three-dimensional diffusive regime. The above results mean that the depolarizing effect of rare magnetic-impurities in the container walls is orders of magnitude larger than a Smoluchowski assumption (to neglect correlations) would imply. This could explain why depolarization is so sensitive to the container's treatment with magnetic fields prior to its use.