Generalized Time-Coarse Graining via an Operator Cumulant Expansion

We introduce a general framework for deriving effective dynamics from arbitrary time-dependent generators, based on a systematic operator cumulant expansion. Unlike traditional approaches, which typically assume periodic or adiabatic driving, our method applies to systems with general time dependencies and is compatible with any dynamics generated by a linear operator -- Hamiltonian or not, quantum or classical, open or closed. This enables modeling of systems exhibiting strong modulation, dissipation, or non-adiabatic effects. Our approach unifies Hamiltonian techniques such as Lie-transform Perturbation Theory (LPT) with averaging-based methods like Time-Coarse Graining (TCG), revealing their structural equivalence through the lens of generalized cumulants. It also clarifies how non-Hamiltonian terms naturally emerge from averaging procedures, even in closed systems. We illustrate the power and flexibility of the method by analyzing a damped, parametrically driven Kapitza pendulum, a system beyond the reach of standard tools, demonstrating how accurate effective equations can be derived across a wide range of regimes.