Effects of sound modes on the VACF in cellular automation fluids

Abstract

The velocity auto correlation function(VACF) of a tagged particle has a fundamental significance in non-equilibrium statistical mechanics. Its time integral defines the self diffusion coefficient through the Green-Kubo relation, and its decay exhibits short time kinetic relaxation, conceivably an intermediate cage effect, and long time hydrodynamic relaxation. Recently Frenkel, van der Hoef and Ernst [1-4] have obtained the VACF in the 2-D FHP-III model and the quasi 3-D FCHC model with remarkable statistical accuracy using lattice gas cellular automata (LGCA). Their results show excellent agreement with the asymptotic long time tails of t-dI2 predicted by mode coupling theory for times larger than 20 x to with to being the mean free time. Mode coupling theory assumes that the long time relaxation to equilibrium can be described through the decay of products of hydrodynamic modes [5, 6]. In particular, the combination of a shear and a self diffusion mode leads to the asymptotic long time tail of the VACF [2, 4]. Obviously, the VACF should approach zero for long times, possibly through power law decay. However, when performing Molecular Dynamics (MD-) simulations on a one-dimensional CA-fluid, the VACF of a tagged particle appeared to approach to a negative constant, as illustrated in Fig. 1 for a system of 500 lattice sites on a line. It was this remarkable observation, suggesting the .existence of some type of conserved quantity, that motivated the present research. Is this a typical one-dimensional effect or could it conceivably also be observed in higher dimensional systems? Our objective here is not only to give a quantitative explanation of this constant anticorrelation, but also to investigate the VACF in the transition region from kinetic relaxation to long time tails, using mode coupling theory. Following Erpenbeck and Wood [7] the mode coupling theory will be extended (i) to include all possible product of pairs of hydrodynamic modes (also those yielding subleading asymptotic time behavior), and (ii) to obtain finite size corrections by