Structure of Stochastic Dynamics near Fixed Points

TL;DR

This paper investigates the structure of stochastic dynamics near fixed points, revealing the existence of a Boltzmann-like stationary distribution that can have non-equilibrium currents and is constructed explicitly using Jordan transformations.

Contribution

It introduces a method to explicitly construct a Boltzmann-like distribution near fixed points, accounting for non-equilibrium currents and non-uniqueness due to nonlinear effects.

Findings

A Boltzmann-like stationary distribution exists near fixed points.

The force can be decomposed into parts satisfying and breaking detailed balance.

Explicit construction of the distribution using Jordan transformation of the force matrix.

Abstract

We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition, but might have instead a divergence-free probability current. In the linear case the force can be split into two parts, one of which gives detailed balance with the diffusive motion, while the other induces cyclic motion on surfaces of constant cost function. Using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.