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

Abstract

Abstract We first introduce the Zwanzig-Kawasaki version of the Generalized
Langevin Equation (GLE) and show as a preamble and under some hy-
pothesis about the relaxation of the fluctuations in the orthogonal sub-
space, 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 Dissi-
pative Particle Dynamics (DPD) for the simulation of complex molecular
systems. This advancement opens up new possibilities for understand-
ing and predicting the behavior of such systems in various scientific and
engineering applications.