Snap, crackle, and pop: This is why the potential of mean force clashes with the fluctuation dissipation relation

Abstract We analyze the non-linear generalized Langevin equation which contains a thermodynamic force. Our method bypasses the projection-operator formalism. We show that even for systems in thermal equilibrium the presence of the thermodynamic force implies that the auto-correlation function of the fluctuating force becomes non-stationary. We further illustrate that a standard coarse-graining procedure that neglects this fact predicts waiting-time distributions incompatible with the original, microscopic process. We conclude that one needs to proceed with care when adding thermodynamic driving forces to the Langevin equation.