The Classical Lattice-Gas Method

1 2 CONTENTS Chapter 1 Introduction Presented is a review of the classical lattice-gas method that deals with an artificial many-body system of particles that has severely discretized microscopic dynamics and that behaves like a viscous Navier-Stokes fluid in the long wavelength hydrodynamic limit. We explain and analytically quantify how the artificial lattice-gas system behaves and we derive a set of criterion that specifies under what conditions it can serve as an appropriate model of a viscous and compressible fluid. Then, we show how the lattice-gas algorithm works using two test models. Finally, we compare the numerical predictions obtained from a variety of different simulations of these two test models to the respective analytical predictions we previously obtained for these models. The resulting numerical and analytical predictions are in good agreement in all cases, but this is only after many failed attempts that were incrementally corrected over time by removing flaws from the derivation of the analytical predictions as well as removing numerical bugs in the implementation of the algorithm and data collection methodology. Therefore, the reason for the consistently good agreement between numerical and analytical predictions is that the derived criteria set has been so sharply delineated that we now know with great accuracy how to initialize the numerical model within a narrow parameter regime where the lattice-gas system is operative. If the initial state of the lattice-gas system is outside this narrow operating regime, the numerical predictions are not at all in agreement with the analytical predictions and the behavior of the long wavelength modes in the system can no longer be classified as hydrodynamical. We have not attempted to catalog any of the non-hydrodynamical behaviors of a classical lattice-gas system. Instead, we have chosen to pursue a narrower goal, which as it turns out is computationally more difficult to pursue, where we run the algorithm only in a parameter regime where it behaves like a fluid. Much information about the details of the microscopic state of a many-body system is not relevant to the hydrodynamic behavior of the many-body system at the macroscopic scale. So in numerically simulating the macroscopic scale fluid behavior, it is possible to arrange for the computer to keep track of many fewer particles than are in the actual system. In a typical lattice-gas simulation of fluid, there are about 10 9 particles. In the simplest type of simulation, all the particles are …