Martingale drift of Langevin dynamics and classical canonical spin statistics -- II

TL;DR

This paper analytically explains how Langevin dynamics with a specific drift function exhibit martingale properties and how the generated histories align with classical spin statistics, revealing new physical insights into martingale theory.

Contribution

It introduces a martingale in the spin functional space to analytically explain the asymptotic spin behavior observed in Langevin dynamics.

Findings

Histories become asymptotically ballistic with orientations following classical spin statistics.

Martingale in spin space provides a new physical perspective on Langevin dynamics.

Analytical explanation of numerical observations from previous work.

Abstract

In the previous paper we have shown analytically that, if the drift function of the d-dimensional Langevin equation is the Langevin function with a properly chosen scale factor, then the evolution of the drift function is a martingale associated with the histories generated by the very Langevin equation. Moreover, we numerically demonstrated that those generated histories from a common initial data become asymptotically ballistic, whose orientations obey the classical canonical spin statistics under the external field corresponding to the initial data. In the present paper we provide with an analytical explanation of the latter numerical finding by introducing a martingale in the spin functional space. In a specific context the present result elucidates a new physical aspect of martingale theory.