On Geometry of Dissipation in Multiscale Dynamics and Thermodynamics

TL;DR

This paper introduces new geometric approaches to multiscale dynamics and thermodynamics, extending kinetic theory, unifying Hamiltonian and gradient dynamics, and incorporating microturbulence within a contact-geometric framework.

Contribution

It presents a novel contact geometric formulation of kinetic theory, a unified geometric framework for Hamiltonian and gradient dynamics, and extends non-equilibrium thermodynamics to include microturbulence.

Findings

Explicit gauge freedom in kinetic theory formulation

Unified geometric framework for Hamiltonian and gradient dynamics

Inclusion of microturbulence in thermodynamic 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.