  We present a detailed derivation and testing of our approach to rescale the
dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y.
Lyubimov et al. J. Chem. Phys. \textbf{132}, 11876, 2010). Starting from the
first-principle Liouville equation and applying the Mori-Zwanzig projection
operator technique, we derive the Generalized Langevin Equations (GLE) for the
coarse-grained representations of the liquid. The chosen slow variables in the
projection operators define the length scale of coarse graining. Each polymer
is represented at two levels of coarse-graining: monomeric as a bead-and-spring
model and molecular as a soft-colloid. In the long-time regime where the
center-of-mass follows Brownian motion and the internal dynamics is completely
relaxed, the two descriptions must be equivalent. By enforcing this formal
relation we derive from the GLEs the analytical rescaling factors to be applied
to dynamical data in the coarse-grained representation to recover the monomeric
description. Change in entropy and change in friction are the two corrections
to be accounted for to compensate the effects of coarse-graining on the polymer
dynamics. The solution of the memory functions in the coarse-grained
representations provides the dynamical rescaling of the friction coefficient.
The calculation of the internal degrees of freedom provides the correction of
the change in entropy due to coarse-graining. The resulting rescaling formalism
is a function of the coarse-grained model and thermodynamic parameters of the
system simulated. The rescaled dynamics obtained from mesoscale simulations of
polyethylene, represented as soft colloidal particles, by applying our
rescaling approach shows a good agreement with data of translational diffusion
measured experimentally and from simulations. The proposed method is used to
predict self-diffusion coefficients of new polyethylene samples.
