Hamilton-Dirac Formulation for Thermodynamic Systems with Reaction and Diffusion

In this paper, we propose a Hamilton-Dirac formulation for non-simple nonequilibrium thermodynamic systems with chemical reactions and diffusion. A key feature of these systems is the degeneracy of their associated Lagrangian function. To address this, we build upon Dirac’s theory of constraints for degenerate Lagrangians and develop a Hamiltonian variational formulation for nonholonomic systems with nonlinear thermodynamic constraints, as well as primary constraints arising from the degeneracy. We introduce the constrained Hamiltonian on the primary constraint set and clarify the underlying geometric structure using Dirac structures. Finally, we illustrate our Hamilton-Dirac formulation with an example of a membrane undergoing matter reaction and diffusion.