Rate of Entropy Production in Stochastic Mechanical Systems

TL;DR

This paper establishes strict bounds on the entropy production rate in various stochastic mechanical systems, linking it to Fisher information and exploring systems with and without damping on different geometric spaces.

Contribution

It introduces new bounds on entropy production rates in stochastic mechanical systems, encompassing systems on Lie groups, with and without damping, and under topological constraints.

Findings

Entropy production rate is closely related to Fisher information.

Bounds on entropy production are derived using classical information theory.

Different equilibrium behaviors are characterized for systems with and without damping.

Abstract

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.