Dirac structures and variational formulation of port-Dirac systems in nonequilibrium thermodynamics

The notion of implicit port-Lagrangian systems for nonholonomic mechanics was proposed in Yoshimura & Marsden (2006a, J. Geom. Phys., 57, 133-156; 2006b, J. Geom. Phys., 57, 209-250; 2006c, Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto) as a Lagrangian analogue of implicit port-Hamiltonian systems. Such port-systems have an interconnection structure with ports through which power is exchanged with the exterior and which can be modeled by Dirac structures. In this paper, we present the notions of implicit port-Lagrangian systems and port-Dirac dynamical systems in nonequilibrium thermodynamics by generalizing the Dirac formulation to the case allowing irreversible processes, both for closed and open systems. Port-Dirac systems in nonequilibrium thermodynamics can be also deduced from a variational formulation of nonequilibrium thermodynamics for closed and open systems introduced in Gay-Balmaz & Yoshimura (2017a, J. Geom. Phys., 111, 169-193; 2018a, Entropy, 163, 1-26). This is a type of Lagrange-d'Alembert principle for the specific class of nonholonomic systems with nonlinear constraints of thermodynamic type, which are associated to the entropy production equation of the system. We illustrate our theory with some examples such as a cylinder-piston with ideal gas, an electric circuit with entropy production due to a resistor and an open piston with heat and matter exchange with the exterior.