  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) 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 explicitly depends on the
initial state of the system, 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
non-linear memory friction contribution is shown to be related to correlations
between the reaction-coordinate velocity and the random force. We present a
numerical method to compute all parameters of our GLE, in particular, the
non-linear memory friction function and the random force distribution, from a
trajectory in reaction coordinate space. We apply our method to 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
significant 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 underlying many-body 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.
