Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems

TL;DR

This paper derives stationary solutions of Liouville equations for a broad class of non-Hamiltonian, dissipative systems, linking their distributions to Hamiltonian-based constraints and identifying special cases like canonical-dissipative systems.

Contribution

It analytically characterizes stationary distributions for non-Hamiltonian systems using a non-holonomic constraint related to phase volume change.

Findings

Distributions are derived analytically for non-Hamiltonian systems.

The class includes systems with constant temperature and canonical-dissipative systems.

Stationary solutions relate phase volume change to non-potential forces.

Abstract

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.