Mathematics Colloquia and Seminars

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Numerical simulations of physical equations of motion always involve a discretization of time. As the time step is increased the discrete-time behavior becomes increasingly different from that of the continuous-time equations of motion. This feature creates a dilemma for any simulation of a dynamical system: use a small time step, resulting in dynamics that resemble continuous-time behavor at the expense of efficiency; or use a large time step that makes the
simulation finish sooner at the expense of meaningful evolution. It is therefore essential to understand the features of different algorithms, such that optimal properties can be chosen for a given set of problems and objectives. We review key basic methods and identify fundamental properties, good and bad, that characterize their influence on the simulated behavior. We are specifically interested in conservation properties for closed systems as well as unavoidable obstacles to complete descriptions of physical properties in discrete time. We then review our work on improvements for systems in thermal equilibrium. The simple derivation of a stochastic Stormer-Verlet algorithm (a thermostat) for the evolution of Langevin equations in a manner that preserves proper configurational
sampling (diffusion and Boltzmann distribution) in discrete time is sketched. The method, which is as simple as conventional Verlet schemes, has been successfully tested on both low-dimensional nonlinear systems as well as more complex ensembles with many degrees of freedom. We then describe and discuss a companion algorithm for controlling pressure in molecular ensembles; i.e., a
barostat for the so-called NPT ensemble. Drawing on the idea of Andersen, we consider a global variable (a virtual piston), which emulates the dynamics of the simulated volume in systems with periodic boundary conditions. However, our description of the dynamics is defined differently from previous work and leads to a very simple set of discrete-time equations that is easily implemented and tested for statistical accuracy in existing MD codes. We sketch the derivation and motivation for our new algorithm, and show favorable comparisons against state-of-the-art algorithms when simulating molecular ensembles at constant pressure and temperature.