gGENERIC: A Variational Framework for Nonequilibrium Thermodynamics via the Entropy Variational Law (EVL)

The GENERIC framework provides a robust structure for nonequilibrium dynamics but  lacks a principled method to select reversible (L) and irreversible (M) brackets. Similarly, f inite-time optimizations minimizing path-averaged reciprocal temperature exist but remain isolated. Here, we introduce the Entropy Variational Law (EVL)—a principle elevating  minimization of (dS/dE)dE to a guiding rule for nonequilibrium evolution. Embedded  in GENERIC, EVL yields gGENERIC, prescribing dynamics via a concrete Poisson–EVL  splitting scheme in which the irreversible step is obtained from a local entropy-variational  quadratic program. At the continuum level, gGENERIC recovers the standard structure of  irreversible thermodynamics and provides a constructive route to bracket selection. Model  problems illustrate how gGENERIC reproduces known optimal heating protocols, standard  thermodynamic laws, and black-hole-inspired entropy balances, suggesting its usefulness as  a flexible building block for entropy-based modelling of nonequilibrium processes in physics  and chemistry.