Numerical Simulation and Coarse-Graining of Large Particle Systems

We discuss the use of combined analysis and simulation to tackle the problem of bridging between the scales in large systems of interacting particles (agents). On the microscopic level we consider several models: Hamiltonian systems such as those that arise in atomistic models for fluids or solids; networks of grain boundaries modeled by evolution PDEs, as well as several stochastic processes. Coarse-graining may involve the passage to large scales in time, space or both. The larger scale models, whether at the mesoscopic or macroscopic levels, involve stochastic processes, stochastic differential equations, ordinary and partial differential equations. The passage between scales is done using two approaches. One involves the construction of a Markov process followed by probabilistic methods combined with simulation. The other follows continuum mechanics arguments combined with simulation. The role of simulation in both approaches is to bridge gaps in the analytical steps, suggesting conjecture about the behavior of certain quantities needed for a complete description on a particular scale.