Fokker-Planck Theory of Nonequilibrium Systems Governed by Hierarchical Dynamics

TL;DR

This paper develops a Fokker-Planck framework for nonequilibrium systems with hierarchical dynamics, specifically analyzing Brownian motion in a fluctuating medium with slowly varying temperature, and introduces a new analytical method for such systems.

Contribution

It formulates a kinetic theory incorporating temperature as a dynamical variable and derives stationary solutions for systems with hierarchical, adiabatic dynamics.

Findings

Stationary distribution is a Maxwellian modulated by temperature fluctuations.

The distribution of temperature fluctuations is determined by the drift term.

A new analytical method is developed for hierarchical stochastic systems.

Abstract

Dynamics of complex systems is often hierarchically organized on different time scales. To understand the physics of such hierarchy, here Brownian motion of a particle moving through a fluctuating medium with slowly varying temperature is studied as an analytically tractable example, and a kinetic theory is formulated for describing the states of the particle. What is peculiar here is that the (inverse) temperature is treated as a dynamical variable. Dynamical hierarchy is introduced in conformity with the adiabatic scheme. Then, a new analytical method is developed to show how the Fokker-Planck equation admits as a stationary solution the Maxwellian distribution modulated by the temperature fluctuations, the distribution of which turns out to be determined by the drift term. A careful comment is also made on so-called superstatistics.