No abstract

We present a lifting procedure for the initialization of molecular dynamics simulations of dense fluids in the context of equation-free computing. As the macroscopic system description we use a smooth representation of the density and velocity profile using a kernel density estimation, of which the dimensionality is reduced via a spline fit. The proposed initialization procedure generates initial atom positions on a grid, as well as initial velocities, such that the desired macroscopic density and velocity profiles are well approximated. We illustrate the approach on a model problem, and point out some inherent caveats in the design of good lifting operators for molecular dynamics.

In the context of multiscale computations, equation-free methods have been developed. In this approach, the evolution of a system is simulated on the macroscopic level while only a microscopic model is explicitly available. To this end, a coarse time stepper for the macroscopic variables can be constructed, based on appropriately initialized microscopic simulations. In this paper, we investigate the initialization of the microscopic simulator using the macroscopic variables only (called lifting in the equation-free framework) when the microscopic model is a molecular dynamics (MD) description of a mono-atomic dense fluid. We assume a macroscopic model to exist in terms of the lowest order velocity moments of the particle distribution (density, velocity and temperature). The major difficulty is to design a lifting operator that accurately reconstructs the physically correct state of the fluid (i.e., the higher order moments) at a reasonable computational cost. We construct a lifting operator, as well as a restriction operator for the reverse mapping. For a simple model problem, we perform a systematic numerical study to assess the time scales on which the lifting errors disappear after reinitialization (healing); we also examine the effects on the simulated macroscopic behavior. The results show that, although in some cases accurate initialization of the higher order moments is not crucial, in general a detailed study of the lifting operator is required.

We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we investigate a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individualbased system. As these parameters might be solution-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation. The results show that for decreasing spatial discretization error, the total error grows rapidly due to the use of estimated coefficients. This effect can be avoided by increasing the quality of the estimates when the spatial discretization decreases. Although the procedure is illustrated for a specific class of multiscale stochastic systems, it is devised so that it can easily be generalized to other stochastic or particle models. Abstract We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we investigate a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individual-based system. As these parameters might be solution-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation. The results show that for decreasing spatial discretization error, the total error grows rapidly due to the use of estimated coefficients. This effect can be avoided by increasing the quality of the estimates when the spatial discretization decreases. Although the procedure is illustrated for a specific class of multiscale stochastic systems, it is devised so that it can easily be generalized to other stochastic or particle models.