In this talk we introduce a model appropriate for the study of stiffness in certain large, highly oscillatory, systems of ordinary differential equations. The model is motivated by the desire to integrate large systems of interacting particles, such as those that arise in molecular dynamics simulations in material science and computational chemistry, without using unduly small time-steps but still retaining some form of accuracy.
A protypical model for statistical mechanics is to consider the motion of a single degree of freedom oscillator (the ``distinguished particle'') coupled by stiff springs to a large number of harmonic oscillators whose natural frequencies span a broad spectrum; the statistical properties of this model are fairly well-understood in the case where randomness is introduced through a measure on the initial data. It is thus a natural test problem within which to study questions concerning the accuracy of numerical simulations of large coupled systems of oscillators. A question of particular importance is to understand whether numerical simulations reproduce the correct statistics for the motion of the distinguished particle when the time-step is large compared to the shortest wavelengths of the coupled harmonic oscillators.
This problem is studied in various distinguished limits described by assumptions made on the natural frequencies of the harmonic oscillators. In particular we study a white noise limit in which the motion of the distinguished particle is governed by the Langevin equation (an Ito stochastic differential equation); this leads naturally to an interesting class of problems concerning approximation theory for ergodic Markov chains. We also study a coloured noise approximation to the white noise limit; this leads to an interesting class of non-Markovian stochastic processes. In both cases computational and analytical results are presented which illustrate some interesting issues concerned with the numerical accuracy of statistical quantities concerned with the distinguished particle.