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

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 nonequilibrium 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.

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

“…Especially, various applications in thermodynamics appeared in numerous papers, to mention Gibbs' book [30], papers by Mrugała et al [63][64][65], and more recent papers [2,4,31,52,[67][68][69]73]. In the latest works by Esen and his collaborators [23,24] the role of contact geometry in statistical mechanics and thermodynamics is deeply studied, including fully geometric formulation of the general equation for non-equilibrium reversible-irreversible coupling. The authors discuss many concepts we also deal with in this paper.…”