Special Seminar in Applied Mathematics

Each particle in a simulation of a system of particles usually represents a huge number of real particles. We present a framework for constructing the dynamics for the so-called coarsened system of simulated particles. We build an approximate solution to the Liouville equation for the original system from the solution of an equation for the phase-space density of a smaller system. We do this with a Markov approximation in a Mori-Zwanzig formalism based on a reference density. We then identify the evolution equation for the reduced phase-space density as the forward Kolmogorov equation of a Markov process. The original system governed by deterministic dynamics is then simulated with the coarsened system governed by this Markov process. Both Monte Carlo (MC) and molecular dynamics (MD) simulations can be view from this framework. More generally, the reduced dynamics can have elements of both MC and MD.