Chapter 2 - Microdynamics: general formalism

Summary

In this chapter, we develop the ‘microdynamic formalism’, which describes the instantaneous microscopic configuration of a lattice gas and its discrete-time evolution. This exact description of the microscopic structure of a lattice gas is the basis for all further theoretical developments, in particular for the prediction of large-scale continuum-like behavior of LGAs.

We first introduce the basic tools and concepts for a general instantaneous description of the microscopic configurations (Section 2.1). The time evolution of the lattice gas is then given in terms of the ‘microdynamic equations’ (Section 2.2). Thereafter, we define microscopic characteristics (e.g. various forms of reversibility), which have a crucial incidence on the macroscopic behavior of the gas, and therefore on its suitability to simulate real physical situations (Section 2.3). The last section is devoted to special rules needed to handle boundary problems (obstacles, particle injections, etc.).

This chapter deals with rather abstract concepts which will find their application in Chapter 3, where lattice gas models are described at the microscopic level.

Basic concepts and notation

The lattice and the velocity vectors

One of the most important features of lattice gases is the underlying Bravais lattice structure which gives a geometrical support to the abstract notion of a cellular automaton. Strictly, a Bravais lattice is by definition infinite. We consider that the cellular automaton only occupies a connected subset ℒ of the D-dimensional underlying Bravais lattice.