Computer simulations of long-time tails: what's new?

Twenty five years ago Alder and Wainwright discovered, by simulation, the 'long-time tails' in the velocity autocorrelation function of a single particle in fluid. Since then, few qualitatively new results on long-time tails have been obtained by computer simulations. However, within the framework of a lattice-gas simulation, we recently developed a technique that makes it possible to 'measure' such velocity autocorrelation functions with an efficiency that is at least a factor 106 higher than can be achieved with conventional techniques. This method opens the way for making comparisons with the predictions of mode-coupling theory of long-time tails, to an accuracy which was hitherto not possible. In this paper we describe this method, and review the results on long-time tails that have been obtained. In particular, we present evidence that the functional form of the long-time tail in a two-dimensional fluid is qualitatively different from the simple power law observed by Alder-Wainwright. Such 'faster-than-t-1' decay is in agreement with the predictions of 'self-consistent' mode-coupling theory.