# Coarse-Grained Monte Carlo Simulations of Arbitrary Continuous Systems

Standard atomistic simulation methods such as Monte Carlo and molecular dynamics have become increasingly powerful as algorithms have improved and computers become faster.  However, the accessible length and time scales of these methods in their basic forms are still firmly grounded at the nanoscale.  As a result, there has been much effort aimed at the development of coarse-grained methods in which some of the system details are removed leading to a more efficient simulation but one in which some of the information is lost.  Key question in coarse-gaining include:

1. How can degrees-of-freedom be removed from the problem representation without affecting the prediction of desired properties?

2. What is the Hamiltonian (i.e. the underlying function that describes the system energy as a function of atomic coordinates) of the coarse-grained system and how can it be derived from the fully resolved atomic system?

In this project, we are developing an on-lattice, coarse-grained Monte Carlo simulation method in which the only input is an inter-particle potential function used in standard full-resolution simulations (e.g. Lennard-Jones potential).  Here, the overall simulation domain is subdivided into an array of sub-volumes, or cells, each of which represent a fixed region of space and can contain a variable number of particles.  The figure below shows the overall nature of the coarse-graining transformation.  In the fully-resolved system (left side panels), a fixed number of particles in a single cell can assume an essentially infinite number of configurations in continuous space, but after coarse-graining, only the number of particles in the cell is retained (right side panels).  As a result, the number of degrees-of-freedom is dramatically reduced.  The essence of the method is then to find a potential function such that the energy change resulting from moves on the coarse-grained system (i.e. moving particles from cell to cell) can be directly informed by the original inter-particle potential function.

Example coarse-grained simulation results are shown in the figure below.  The top row shows three equilibrium configurations from fully-resolved Lennard-Jones (LJ)Monte Carlosimulations of a vapor-liquid system.  Each of these simulations was initiated with a homogeneous distribution of LJ atoms at a fixed density.  All three densities correspond to a liquid-vapor mixture at (or at least near) equilibrium, although the liquid fraction increases from left to right.  The lower row shows the same three simulations using coarse-grainedMonte Carlo.  The color of each cell indicates the number of particles – blue corresponds to low particle density, yellow/orange to medium, and red to high.  The overall vapor and liquid phase densities are in excellent agreement with the fully resolved predictions; in fact, the entire phase diagram can be reproduced.  Note that the coarse-grained simulation cells effectively contain several million atoms and reach equilibrium in a fraction of the time that the (much smaller) fully resolved calculations do.

(Project is in collaboration with Prof. W. D. Seider at the University of Pennsylvania.)