Martingale drift of Langevin dynamics and classical canonical spin statistics

TL;DR

This paper explores the martingale properties of Langevin dynamics, revealing that the drift amplitude corresponds to the classical Langevin function and that the long-term behavior aligns with classical Heisenberg spin statistics.

Contribution

It establishes a novel connection between martingale properties of Langevin processes and classical spin response functions, providing new insights into stochastic process symmetries.

Findings

Drift amplitude equals the Langevin function.

Asymptotic behavior of the process matches classical spin statistics.

Martingale property characterizes the unbiased nature of the process.

Abstract

The martingale characterizes a kind of fairness or unbiased nature of the stochastic process which is associated with another stochastic process. If $x_t$ evolves according to the Langevin equation whose mean drift is $a_t$ as function of $x_t,$ and that $a_t$ as induced stochastic process is martingale in turn associated with the former process, then we show that the amplitude of $a_t$ is the Langevin function, which is originally the canonical response of a single classical Heisenberg spin under static field. Furthermore, the asymptotic limit of $x_t/t$ obeys the ensemble statistics of such Heisenberg spin.