Phase field crystals as a coarse-graining in time of molecular dynamics

Phase field crystals (PFC) are a tool for simulating materials at the atomic level. They combine the small length scale resolution of molecular dynamics (MD) with the ability to simulate dynamics on mesoscopic time scales. We show how PFC can be interpreted as the result of applying coarse-graining in time to the microscopic density field of molecular-dynamics simulations. We take the form of the free energy for the phase field from the classical density functional theory of inhomogeneous liquids and then choose coefficients to match the structure factor of the time coarse-grained microscopic density field. As an example, we show how to construct a PFC free energy for Weber and Stillinger's two-dimensional square crystal potential which models a system of proteins suspended in a membrane