Mori-Zwanzig projection operator formalism: Generalized Langevin equation dynamics of a classical system perturbed by an external generalized potential and far from equilibrium

A new derivation of the generalized Langevin equation (GLE) of motion is presented that exactly describes the reduced dynamics of a particle system acted upon by a time-dependent force field with a generalized (velocity-dependent) potential of general form and arbitrarily far from equilibrium. The derivation is carried out using the Mori-Zwanzig formalism with the stationary Zwanzig projection operator, which is defined on the initial phase variables to exactly project the microscopic Hamiltonian equations of motion. The first major difference with the GLE for the systems perturbed by velocity-independent potentials is the presence of several types of memory functions that rely on the various correlations of projected (fluctuating) forces and velocities. The second difference is a nontrivial GLE dynamics for the coarse-grained positional coordinates, which arises due to the differences in the canonical and kinetic momenta. Similarities include: The projected force remains a two-time-scale process with scales due to the interparticle interactions and the time-dependence of the external forces; the memory functions and the projected forces are related via the explicit time-dependent fluctuation-dissipation relation of the second kind; and the dissipative memory terms are linear functionals of the gradients of the Boltzmann entropy of the irrelevant subsystem that agree with Onsager's phenomenological equations. The theory is applied to a particle-based coarse-graining (via clustering of the microscopic particles). The theoretical presentation is illustrated with an example of a system of charged particles coupled to an electromagnetic field. The GLEs and workflow presented here can be readily used as a starting point to rigorously formulate microscopically informed multiscale treatments (e.g., using dissipative particle dynamics) for a variety of phenomena in particle systems perturbed by velocity-dependent fields, such as the external Lorentz, centrifugal, or Coriolis forces and in nonequilibrium.