# Dirac structures in nonequilibrium thermodynamics

## TL;DR

This paper demonstrates that the evolution equations in nonequilibrium thermodynamics can be intrinsically formulated using Dirac structures, extending geometric mechanics to include irreversible processes through a unified geometric framework.

## Contribution

It introduces a novel geometric formulation of nonequilibrium thermodynamics using Dirac structures, generalizing the mechanics framework to incorporate irreversible processes.

## Key findings

- Dirac structures naturally describe nonequilibrium thermodynamics evolution equations
- In absence of irreversibility, structures reduce to canonical symplectic forms
- Provides a unified geometric framework extending mechanics to thermodynamics

## Abstract

Dirac structures are geometric objects that generalize both Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems. In this paper, we show that the evolution equa- tions for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, both on the Lagrangian and the Hamiltonian settings. In absence of irreversible processes these Dirac structures reduce to canonical Dirac structures associated to canonical symplectic forms on phase spaces. Our geometric formulation of nonequilibrium thermodynamic thus consistently extends the geometric formulation of mechanics, to which it reduces in absence of irreversible processes. The Dirac structures are associated to the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2016a,b] and are induced from a nonlinear nonholonomic constraint given by the expression of the entropy production of the system.