Dirac structures in nonequilibrium thermodynamics for simple open systems

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 with entropy production. We overcome this issue by working with the geometric setting of time-dependent nonholonomic mechanics. We define two types of Dirac dynamical systems for the nonequilibrium thermodynamics of open systems, based either on the generalized energy or the Lagrangian. The variational formulations associated with the Dirac dynamical systems are also presented.