On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective

TL;DR

This paper develops a comprehensive geometric framework for GENERIC, unifying Hamiltonian and gradient dynamics in non-equilibrium thermodynamics through cotangent lifts and contact geometry, clarifying its geometric structure and reduction methods.

Contribution

It introduces a geometric formulation of GENERIC using cotangent lifts and contact geometry, revealing its structure and providing new reduction techniques.

Findings

Cotangent lifts unify Hamiltonian and gradient dynamics.

Lifted vector fields can be split into holonomic and vertical parts.

Contact geometry explicitly incorporates the second law of thermodynamics.

Abstract

This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure, due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.