M.A. van der Hoef

Tagged particle diffusion in 3D lattice gas cellular automata

Abstract We report simulations of tagged particle diffusion in three-dimensional lattice gas cellular automata (LGCA). In particular we looked at the decay of the velocity autocorrelation function (VACF) using a new technique that is about a million times more efficient than the conventional techniques. For longer times the simulations clearly show the algebraic t − 3 2 tail of the VACF. We compare the observed long-time tail with the predictions of mode-coupling theory. In three dimensions, the amplitude of this tail is found to agree within the (small) statistical error with these predictions.

Velocity autocorrelation function in a four-dimensional lattice gas

We report simulations of the velocity autocorrelation function (VACF) of a tagged particle in a four-dimensional lattice gas cellular automaton (LGCA). We observe a hydrodynamic tail in the VACF, which decays as t-2, in agreement with the theoretical predictions. However, in a quantitative comparison, the simulations show that mode-coupling theory underestimates the amplitude of the hydrodynamic tail by (15 ÷ 60)%. The artificial correlations, previously observed in the projected three-dimensional lattice gas model, are found to be absent in this truly 4D model.

A Test of Mode-Coupling Theory

In the history of the kinetic theory of fluids, 1969–1970 was a crucial year. In that year Alder and Wainwright [2] published a paper in which they demonstrated the breakdown of the ‘Molecular Chaos’ assumption. The Molecular Chaos assumption, originally introduced by Boltzmann as the ‘Stoszahlansatz’, states that the collisions experienced by a molecule in a fluid are uncorrelated. One consequence of this assumption is that the velocity autocorrelation function (VACF) of a tagged particle in fluid should decay exponentially. What Alder and Wainwright found is that the VACF of a particle in a moderately dense fluid of hard spheres or hard disks does not decay exponentially but algebraically. These algebraic long-time tails are the consequence of coupling between particle diffusion and shear modes in the fluid.