Liouville's Theorem and the canonical measure for nonconservative systems from contact geometry

TL;DR

This paper extends statistical mechanics to nonconservative systems using contact geometry, deriving generalized Hamilton's equations, Liouville's theorem, and invariant measures with power law distributions.

Contribution

It introduces a contact geometric framework for nonconservative systems, generalizing key concepts of statistical mechanics beyond conservative dynamics.

Findings

Derived generalized Hamilton's equations for contact systems.

Proved Liouville's theorem in the contact geometry context.

Identified invariant measures with power law density distributions.

Abstract

Standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton's equations, Liouville's theorem and to find the canonical invariant measure. In this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry. In particular, we find out the generalized Hamilton's equations, Liouville's theorem and the microcanonical and canonical measures invariant under the contact flow. Remarkably, the latter measure has a power law density distribution with respect to the standard contact volume form. Finally, we argue on the several possible applications of our results.