We consider the Generalized Langevin Equations of motion describing exactly the particle-based coarse-grained (CG) dynamics in the classical microscopic ensemble that were derived recently within the Mori-Zwanzig formalism based on new projection operators. The fundamental difference between the new family of projection operators and the standard Zwanzig projection operator used in the past to derive CG equations of motion is that the new operators average out the explicit irrelevant trajectories leading to the possibility of solving the stochastic dynamics exactly. We present the formalism based on the new projection operators to compute the stochastic dynamics exactly for the microscopic system in equilibrium. The resulting expression for the random force is in the form of a “generalized additive fluctuating force” describing the departure of the generalized microscopic force associated with the CG coordinate from its projection. Starting with this key expression we formulate a new exact formula for the memory function in terms of microscopic and CG conservative forces. We conclude by studying two independent limiting cases of practical importance: the Markov limit (vanishing correlations of random force) and the limit of weak dependence of the memory function on the particle momenta. We present computationally affordable expressions which can be efficiently evaluated from standard molecular dynamics simulations. The expressions for the memory function correctly accounts for the momentum dependence in the lowest second order and indicates that such a dependence may be important in the CG dynamics approaching the Markov limit. In a case of CG dynamics with a weak dependence of the memory effects on the particle momenta, the expression for the memory function presented in this work is applicable to non-Markov systems. A numerical application is presented for a molecular liquid (nitromethane). With the new formalism we do not observe plateau problem if the friction tensors for dissipative particle dynamics (DPD) are computed using the Green-Kubo relation. Our formalism provides a consistent bottom-up route for hierarchical parameterization of DPD models from atomistic simulations.

Microscopic derivation of particle-based coarse-grained dynamics: Exact expressions for dissipative interactions

255th National Meeting (2018)

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