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We present a detailed derivation and testing of our approach to rescale thedynamics of mesoscale simulations of coarse-grained polymer melts (I. Y.Lyubimov et al. J. Chem. Phys. \textbf{132}, 11876, 2010). Starting from thefirst-principle Liouville equation and applying the Mori-Zwanzig projectionoperator technique, we derive the Generalized Langevin Equations (GLE) for thecoarse-grained representations of the liquid. The chosen slow variables in theprojection operators define the length scale of coarse graining. Each polymeris represented at two levels of coarse-graining: monomeric as a bead-and-springmodel and molecular as a soft-colloid. In the long-time regime where thecenter-of-mass follows Brownian motion and the internal dynamics is completelyrelaxed, the two descriptions must be equivalent. By enforcing this formalrelation we derive from the GLEs the analytical rescaling factors to be appliedto dynamical data in the coarse-grained representation to recover the monomericdescription. Change in entropy and change in friction are the two correctionsto be accounted for to compensate the effects of coarse-graining on the polymerdynamics. The solution of the memory functions in the coarse-grainedrepresentations provides the dynamical rescaling of the friction coefficient.The calculation of the internal degrees of freedom provides the correction ofthe change in entropy due to coarse-graining. The resulting rescaling formalismis a function of the coarse-grained model and thermodynamic parameters of thesystem simulated. The rescaled dynamics obtained from mesoscale simulations ofpolyethylene, represented as soft colloidal particles, by applying ourrescaling approach shows a good agreement with data of translational diffusionmeasured experimentally and from simulations. The proposed method is used topredict self-diffusion coefficients of new polyethylene samples.

A First Principle Approach to Rescale the Dynamics of Simulated Coarse-Grained Macromolecular Liquids