Dynamical symmetries of Markov processes with multiplicative white noise

TL;DR

This paper uncovers hidden dynamical symmetries in Markov processes with multiplicative white noise, providing new insights into fluctuation relations, simplifying analysis, and clarifying misconceptions in stochastic process literature.

Contribution

It identifies two dynamical symmetries in multiplicative white-noise Markov processes, linking them to fluctuation theorems and correlation-response relations, and clarifies discretisation effects.

Findings

Two hidden symmetries in generating functionals.

Equilibrium symmetry leads to fluctuation-dissipation relations.

Out-of-equilibrium symmetry yields fluctuation theorems.

Abstract

We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the Landau-Lifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.