Statistical mechanics of the GENERIC framework under external forcing

The General Equation for Non-Equilibrium Reversible Irreversible Coupling (generic) framework provides a thermodynamically consistent approach to describe the evolution of coarse-grained variables. This framework states that Markovian dynamic equations governing the evolution of coarse-grained variables have a universal structure that ensures energy conservation (first law) and entropy increase (second law). However, the presence of external time-dependent forces can break the energy conservation law, requiring modifications to the framework's structure. To address this issue, we start from a rigorous and exact transport equation for the average of a set of coarse-grained variables derived from a projection operator technique in the presence of external forces. Under the Markovian approximation, this approach provides the statistical mechanics underpinning of the generic framework under external forcing conditions. By doing so, we can account for the effects of external forcing on the system's evolution while ensuring thermodynamic consistency.