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Generalized Langevin equations (GLEs) can be systematically derived viadimensional reduction from high-dimensional microscopic systems. For linearmodels the derivation can either be based on projection operator techniquessuch as the Mori-Zwanzig (MZ) formalism or by "integrating out" the bathdegrees of freedom. Based on exact analytical results we show that both routescan lead to fundamentally different GLEs and that the origin of thesedifferences is based inherently on the non-equilibrium nature of themicroscopic stochastic model. The most important conceptional differencebetween the two routes is that the MZ result intrinsically fulfills thegeneralized second fluctuation-dissipation theorem while the integration resultcan lead to its violation. We supplement our theoretical findings withnumerical and simulation results for two popular non-equilibrium systems:Time-delayed feedback control and the active Ornstein-Uhlenbeck process.

Non-Markovian systems out of equilibrium: Exact results for two routes of coarse graining