  In this paper, we introduce the notion of port-Lagrangian systems in
nonequilibrium thermodynamics, which is constructed by generalizing the notion
of port-Lagrangian systems for nonholonomic mechanics proposed in Yoshimura and
Marsden [2006c], where the notion of interconnections is described in terms of
Dirac structures. The notion of port-Lagrangian systems in nonequilibrium
thermodynamics is deduced from the variational formulation of nonequilibrium
thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,2017b]. It is a
type of Lagrange-d'Alembert principle associated to a specific class of
nonlinear nonholonomic constraints, called phenomenological constraints, which
are associated to the entropy production equation of the system. To these
phenomenological constraints are systematically associated variational
constraints, which need to be imposed on the variations considered in the
principle. In this paper, by specifically focusing on the cases of simple
thermodynamic systems with constraints, we show how the interconnections in
thermodynamics can be also described by Dirac structures on the Pontryagin
bundle as well as on the cotangent bundle of the thermodynamic configuration
space. Each of these Dirac structures is induced from the variational
constraint. Furthermore, the variational structure associated to this Dirac
formulation is presented in the context of the Lagrange-d'Alembert-Pontryagin
principle. We illustrate our theory with some examples such as a
cylinder-piston with ideal gas as well as an LCR circuit with entropy
production due to a resistor.
