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We introduce a hybrid projection scheme that combines linear Mori projectionand conditional Zwanzig projection techniques and use it to derive aGeneralized Langevin Equation (GLE) for a general interacting many-body system.The resulting GLE includes i) explicitly the potential of mean force (PMF) thatdescribes the equilibrium distribution of the system in the chosen space ofreaction coordinates, ii) a random force term that explicitly depends on theinitial state of the system, and iii) a memory friction contribution thatsplits into two parts: a part that is linear in the past reaction-coordinatevelocity and a part that is in general non-linear in the past reactioncoordinates but does not depend on velocities. Our hybrid scheme thus combinesall desirable properties of the Zwanzig and Mori projection schemes. Thenon-linear memory friction contribution is shown to be related to correlationsbetween the reaction-coordinate velocity and the random force. We present anumerical method to compute all parameters of our GLE, in particular, thenon-linear memory friction function and the random force distribution, from atrajectory in reaction coordinate space. We apply our method to thedihedral-angle dynamics of a butane molecule in water obtained from atomisticmolecular dynamics simulations. For this example, we demonstrate thatnon-linear memory friction is present and that the random force exhibitssignificant non-Gaussian corrections. We also present the derivation of the GLEfor multidimensional reaction coordinates that are general functions of allpositions in the phase space of the underlying many-body system; thiscorresponds to a systematic coarse-graining procedure that preserves not onlythe correct equilibrium behavior but also the correct dynamics of thecoarse-grained system.

Generalized Langevin Equation with a Non-Linear Potential of Mean Force and Non-Linear Memory Friction From a Hybrid Projection Scheme