# Variational discretization of the nonequilibrium thermodynamics of simple systems

## TL;DR

This paper introduces variational integrators for modeling the nonequilibrium thermodynamics of simple closed systems, extending Lagrangian mechanics to include irreversible processes with structure-preserving properties.

## Contribution

It develops a novel discretization method for nonequilibrium thermodynamics based on variational principles, extending existing integrators to irreversible systems.

## Key findings

- Discrete flow preserves a structure analogous to symplecticity.
- The scheme demonstrates stability and structure preservation in simulations.
- Regularity conditions ensure the existence of discrete flows.

## Abstract

In this paper, we develop variational integrators for the nonequilibrium thermodynamics of simple closed systems. These integrators are obtained by a discretization of the Lagrangian variational formulation of nonequilibrium thermodynamics developed in \cite{GBYo2016a}, and thus extend the variational integrators of Lagrangian mechanics, to include irreversible processes. In the continuous setting, we derive the structure preserving property of the flow of such systems. This property is an extension of the symplectic property of the flow of the Euler-Lagrange equations. In the discrete setting, we show that the discrete flow solution of our numerical scheme verifies a discrete version of this property. We also present the regularity conditions which ensure the existence of the discrete flow. We finally illustrate our discrete variational schemes with the implementation of an example of a simple and closed system.