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

Abstract

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a Generalized Langevin Equation (GLE) from the Liouville equation for a general interacting many-body system. The resulting GLE includes i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, ii) a random force term that is a function of the initial state of the system only, and iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general non-linear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear friction contribution is caused by correlations between the reaction-coordinate velocity and the random force. We present an algorithm to compute all parameters of the GLE, in particular the non-linear friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example we demonstrate that non-linear memory friction is present and that the random force exhibits non-negligible non- Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase space of the original system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.