Irreversible dynamics on Poisson manifolds

TL;DR

This paper introduces a geometric method to construct irreversible dynamics on Poisson manifolds using deformation theory, satisfying key principles of thermodynamics without extra metric structures.

Contribution

It provides a novel construction of symmetric brackets from Poisson structures' deformation, applicable to classical and infinite-dimensional systems, advancing the understanding of irreversible processes.

Findings

Constructed symmetric brackets from Poisson structures' deformation.

Applied the method to Lie algebra duals of SE(2) and SGal(3).

Demonstrated relevance to classical mechanics and control theory.

Abstract

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz-Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples, the duals of the Lie algebras of the special Euclidean group SE(2) and the Galilei group SGal(3). These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics. Our framework naturally extends to infinite-dimensional settings, which are discussed as directions for future work.