Viscosity of a Lattice Gas

The shear viscosity of a lattice gas can be derived in th e Boltzmann approximat ion from a straightfor ward analysis of th e numerical algorithm. Thi s computat ion is presented first in the case of the Friech-Hasslacher-Pome au two-dimensional triangular lattice. It is then generalized to a regular lat t ice of arbitrary dimension, shape, and collision rules with appropriate symmetries . The viscosity is shown to be positive. A practical recip e is given for choosing collision rules so as to minimize th e viscosity. 1. Int roduction 1.1 Goal For computational efficiency, the collision rules in a lattice gas automaton should be chosen so as to make the shear viscosity as small as possible [1]. For a given set of rules, the viscosity can be estimated through numerical simulations [2,3]. It would be much more convenient, however, to have an explicit formula through which the viscos ity could be computed directly from the lattice rules . Here, I present such a formula , which is applicable when the following conditions are sat isfied: 1. there is a sing le population of particles (all velocities have the same modulus); 2. the lattice and the collision ru les are "sufficiently symmetrical", in a sense which will be made more precise below (section 3); 3. the Boltzmann approximation is valid (the probabilities of arrival of particles at a node from different direct ions can be assumed independent) ; 4. the system is not far from isotropic equilibrium (low Mach number) j 5. the on ly quantities conserved by collisions are the number of particles and the momentum; and 6. the collisions satisfy semi-detailed balancing . It can be shown from the formu la that the viscosity is always positive. Moreover, from the structure of the formula, one can derive a simple rule for the optimization of ind ividual collisions . © 1987 Complex Systems Publications, Inc.