Classical Canonical Distribution for Dissipative Systems

TL;DR

This paper derives the canonical distribution as a stationary solution for dissipative classical systems, showing conditions under which dissipative forces lead to Gibbs-like stationary states, exemplified by a damped oscillator.

Contribution

It provides a derivation of the canonical distribution for dissipative systems and identifies simple conditions for non-potential forces to produce Gibbs-like stationary states.

Findings

Canonical distribution can be derived as a stationary solution of the Liouville equation for dissipative systems.

Stationary states resemble canonical Gibbs distributions under specific force conditions.

Example of a linear oscillator with friction illustrates the theoretical results.

Abstract

In this paper we derive the canonical distribution as a stationary solution of the Liouville equation for the classical dissipative system. Dissipative classical systems can have stationary states look like canonical Gibbs distributions. The condition for non-potential forces which leads to this stationary solution is very simple: the power of the non-potential forces must be directly proportional to the velocity of the Gibbs phase (phase entropy density) change. The example of the canonical distribution for a linear oscillator with friction is considered.