Constraints in nonequilibrium thermodynamics: General framework and application to multicomponent diffusion

We elaborate how holonomic constraints can be incorporated into the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Dirac's ideas for constructing constrained Poisson brackets are extended to dissipative brackets. The construction is presented such that it can be put into practice most readily. We illustrate the procedure by developing a symmetric thermodynamic description of diffusion in multicomponent systems and, as a further example, we impose an incompressibility constraint. As a consequence of its more elaborate and restrictive structure, GENERIC removes the ambiguities occurring in the classical thermodynamics of irreversible processes when one works with redundant variables.