Nonlinear Mori–Zwanzig theory and quadratic coarse-grained coordinates for complex molecular systems

Abstract

We first introduce the Zwanzig–Kawasaki version of the generalized Langevin equation and show as a preamble and under some hypothesis about the relaxation of the fluctuations in the orthogonal subspace, that the commonly used term for the Markovian approximation of the dissipation is rigorously vanishing, necessitating the use of the next-order term, in an integral series we introduce. Independently, we provide thereafter a comprehensive description of complex coarse-grained molecules which, in addition to the classical positions and momenta of their centers of mass, encompasses their shapes, angular momenta and internal energies. The dynamics of these quantities is then derived as the coarse-grained forces, torques, microscopic stresses, energy transfers, from the coarse-grained potential built with their Berne-like anisotropic interactions. By incorporating exhaustively the quadratic combinations of the atomic degrees of freedom, this novel approach enriches considerably the dynamics at the coarse-grained level and could serve as a foundation for developing numerical models more holistic and accurate than dissipative particle dynamics for the simulation of complex molecular systems. This advancement opens up new possibilities for understanding and predicting the behavior of such systems in various scientific and engineering applications.