On Geometry of Dissipation in Multiscale Dynamics and Thermodynamics

TL;DR

The paper develops a geometric program to understand dissipation in multiscale dynamics by embedding kinetic theory and nonequilibrium thermodynamics in contact geometry. It introduces a gauge-invariant contact kinetic theory that includes phase-space volume as a dynamical variable, a graph-space realization of GENERIC within the geometric Hamilton–Jacobi framework, and a geometric extension of non-equilibrium thermodynamics that incorporates microturbulence. The approach yields both momentum and density formulations of kinetic dynamics and unifies Hamiltonian and dissipative evolution under a single Jacobi/contact structure. The results provide a principled, coordinate-free framework for analyzing irreversible processes in multiscale systems and suggest new directions in Lie algebroid formulations and entropy production modeling.

Abstract

This manuscript introduces novel approaches to three phenomena. First, we extend the algebraic formulation of kinetic theory within the contact framework by making explicit the gauge freedom, thereby obtaining a formulation in which the phase-space volume itself becomes an additional dynamical variable. Second, we develop a new and simpler geometric formulation of the GENERIC framework, unifying Hamiltonian and gradient dynamics in a contact-geometric setting. This is realized within a specifically constructed graph space, which naturally emerges as an intermediate structure in the geometric Hamilton-Jacobi framework. Finally, we formulate a geometric extension of non-equilibrium thermodynamics in the setting of geometric Hamilton-Jacobi theory, allowing for the inclusion of microturbulence - a key feature of complex dynamical systems.