Dirac Structures in Nonequilbrium Thermodynamics

Abstract

In this paper, we show that the evolution equations for nonequilibrium thermodynamics can be formulated in terms of Dirac structures on the Pontryagin bundle \(\mathsf {P} =T \mathsf {Q}\oplus T ^* \mathsf {Q} \), where \(\mathsf {Q}=Q \times \mathbb {R}\) denotes the thermodynamic configuration manifold. In particular, we extend the use of Dirac structures from the case of linear nonholonomic constraints to the case of nonlinear nonholonomic constraints. Such a nonlinear constraint comes from the entropy production associated with irreversible processes in nonequilibrium thermodynamics. We also develop the induced Dirac structure on \(N=T^*Q \times \mathbb {R} \) and the associated Lagrange-Dirac and Hamilton-Dirac dynamical formulations.