# Dirac structures in nonequilibrium thermodynamics for simple open systems

## TL;DR

This paper introduces a geometric framework using Dirac structures to model the dynamics of simple open thermodynamic systems exchanging heat and matter, extending the geometric approach to nonequilibrium thermodynamics.

## Contribution

It develops a novel formulation of open thermodynamic systems using Dirac structures within a time-dependent nonholonomic mechanics framework, addressing explicit time dependence issues.

## Key findings

- Formulation of Dirac dynamical systems for open thermodynamics
- Introduction of variational principles for these Dirac systems
- Application to simple systems with entropy as the state variable

## Abstract

Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evolution equations for open thermodynamic systems, i.e., systems exchanging heat and matter with the exterior, admit an intrinsic formulation in terms of Dirac structures. We focus on simple systems, in which the thermodynamic state is described by a single entropy variable. A main difficulty compared to the case of closed systems lies in the explicit time dependence of the constraint associated to the entropy production. We overcome this issue by working with the geometric setting of time-dependent nonholonomic mechanics. We define three type of Dirac dynamical systems for the nonequilibrium thermodynamics of open systems, based either on the generalized energy, the Lagrangian, or the Hamiltonian. The variational formulations associated to the Dirac systems formulations are also presented.