The geometry of some thermodynamic systems

TL;DR

This paper develops a geometric framework for modeling thermodynamic systems, incorporating evolution vector fields, skew-symmetric brackets, and Lagrangian-Hamiltonian formalisms to describe their dynamics and interactions.

Contribution

It introduces a novel geometric approach to thermodynamics using evolution vector fields and brackets, extending to composite systems with heat exchange.

Findings

Formulation of thermodynamic evolution via vector fields.

Integration of discrete gradient methods with geometric structures.

Application to composite systems with multiple thermal variables.

Abstract

In this article, we continue the program started in our previous article of exploring an important class of thermodynamic systems from a geometric point of view. In order to model the time evolution of systems verifying the two laws of thermodynamics, we show that the notion of evolution vector field is adequate to appropriately describe such systems. Our formulation naturally arises from the introduction of a skew-symmetric bracket to which numerical methods based on discrete gradients fit nicely. Moreover, we study the corresponding Lagrangian and Hamiltonian formalism, discussing the fundamental principles from which the equations are derived. An important class of systems that is naturally covered by our formalism are composed thermodynamic systems, which are described by at least two thermal variables and exchange heat between its components.}.