Our discussion of lattice-gas models now takes a qualitative turn. We continue to study fluid mixtures as in the previous chapter, but now they will exhibit some surprising behavior—they won't like to mix!

This change in direction also steers us towards the heart of this book: models for complex hydrodynamics. The particular kind of complexity we introduce in this chapter relates to interfaces in immiscible fluids such as one might find in a mixture of oil and water. We are all familiar with the kind of bubbly complexity that that can entail. So it seems all the more remarkable that only a revised set of collision rules are needed to simulate it with lattice gases. Indeed, the models of immiscible fluids that we shall introduce are so close to the models of the previous chapters that we call them immiscible lattice gases.

This chapter, an introduction to immiscible lattice-gas mixtures, is limited to a discussion of two-dimensional models. In the next chapter, we introduce a lattice-Boltzmann method that is the “Boltzmann equivalent” of the immiscible lattice gas. That then sets the stage for our discussion of three-dimensional immiscible lattice gases in Chapter 11.

Color-dependent collisions

In the miscible lattice gases of the previous chapter, the collision rules were independent of color. The diffusive behavior derived instead from the redistribution of color after generic colorblind collisions were performed. Aside from some diffusion, the color simply went with the flow.

In this chapter we give a full derivation of the Navier-Stokes equation for the lattice gas. The first step, the Boltzmann approximation, is an approximation of the exact Liouville dynamics. The Boltzmann approximation should not be confused with the lattice-Boltzmann method of Chapter 6 and Chapter 7, but results that we obtain here for the Boltzmann approximation and the Navier-Stokes equations are also useful for the Boltzmann method. One of these results is the H-theorem for lattice gases. In this chapter we also reopen the tricky issue of the spurious invariants of lattice-gas or Boltzmann dynamics. We specifically discuss non-uniform global linear invariants for which, unlike the general nonlinear invariants, some theoretical results are known. Among them we find the staggered momentum invariants. We discuss their effect on hydrodynamics, which leads us to corrections to the Euler equation of Chapter 2.

General Boolean dynamics

It is useful to express the Boolean dynamics as a sequence of Boolean calculations, as we did in Section 2.6. In this section we shall denote by s or n local configurations. We further define a field of “rate bits”, which are random, Boolean variables, defined independently on each site and denoted by ass′(x, t). They are equal to one with probability 〈ass′〉 = A(s, s′). Thus if the pre-collision configuration is n the post-collision configuration is that single s′ for which ans′ = 1.

Our objectives for this chapter are twofold. First, we review some elementary aspects of fluid mechanics. We include in that discussion a classical derivation of the Navier-Stokes equations from the conservation of mass and momentum in a continuum fluid. We then discuss the analogous conservation relations in a lattice gas. Finally, we briefly describe the derivation of hydrodynamic equations for the lattice gas, but defer our first detailed discussion of this subject to the following chapter.

Molecular dynamics versus continuum mechanics

The study of fluids typically proceeds in either of two ways. Either one begins at the microscopic scale of molecular interactions, or one assumes that at a particular macroscopic scale a fluid may be described as a smoothly varying continuum. The latter approach allows us to write conservation equations in the form of partial-differential equations. Before we do so, however, it is worthwhile to recall the basis of such a point of view.

The macroscopic description of fluids corresponds to our everyday experience of flows. Figure 2.1 shows that a flow may have several characteristic length scales li. These lengths scales may be related either to geometric properties of the flow such as channel width or the diameter of obstacles or to intrinsic properties such as the size of vortical structures. The smallest of these length scales will be called Lhydro.

Non-dissipative, inviscid, compressible flow

We consider a simple fluid such as air or water. This fluid is described by a number of thermodynamical fields, such as pressure, density, etc., as well as a velocity field u. A variable is called specific when it gives a quantity per unit mass. For instance, let E be the internal energy of a finite volume V of fluid. Let M = ρV be the mass of this volume of fluid. Then e = E/M is the specific internal energy. Table C.1 lists all the thermodynamic variables used in this appendix.

In addition to thermodynamic variables there are variables describing the external actions on the fluid. The effect of gravity, for instance may be represented by the acceleration f = g. The heating rate per unit mass q represents sources of heat, for instance from radiation. There may be heat and momentum exchanges inside the fluid, by heat conduction or viscous forces. These are not taken into account in the non-dissipative description.

For a one-component gas, there are only two independent thermodynamic variables. As a special choice, one may choose ρ, T as independent variables and express all quantities such as p, e, h, etc., in terms of ρ and T, but other choices are possible as well. These relations are linked through equations of state. For instance p = p(ρ, T) is the standard form for an equation of state.

We now provide our first detailed derivation of the hydrodynamics of lattice gases. To keep matters from becoming unnecessarily complicated, we mostly restrict the discussion in this chapter to two-dimensional (2D) models. We begin with the simplest possible 2D model on a square lattice. We then repeat the calculation for the hexagonal lattice model. The principal result of this chapter is the derivation of the Euler equation of both models. This equation has the form already indicated in Chapter 2 but here the unknown coefficients are explicitly calculated. For future use we also include a general calculation in arbitrary dimension D and with an arbitrary number of rest particles.

The hydrodynamic behavior that we thus find at the macroscopic scale is a consequence of the existence of a kind of thermodynamic equilibrium. This equilibrium state is described by the Fermi-Dirac distribution of statistical mechanics. How this distribution arises is described in detail in Chapters 14 and 15. In this chapter we give a simpler derivation of some properties of equilibrium, which are sufficient to obtain the Euler equation.

Homogeneous equilibrium distribution on the square lattice

Our first task is to calculate 〈ni〉, the average value of the Boolean variable ni introduced in Section 2.6. Repeated applications of the rules of propagation and collision in the lattice gas cause these average particle populations to quickly reach an equilibrium state regardless of initial conditions.

We believe that the computer simulation of physics should retain the elegance of physics itself. This book is about one approach towards this objective: lattice-gas cellular automata models of fluids.

Cellular automata are fully discrete models of physical and other systems. Lattice gases are just what the name says: a model of a gas on a grid. Thus lattice-gas cellular automata are a special kind of gas in which identical particles hop from site to site on a lattice at each tick of a clock. When particles meet they collide, but they always stay on the grid and appropriate physical quantities are always conserved.

Our subject is interesting because it provides a new way of thinking about the simulation of fluids. It also provides an instructive link between the microscopic world of molecular dynamics and the macroscopic world of fluid mechanics. Lastly, it allows us to create new computational tools that can be usefully applied to solve certain problems.

There are, broadly speaking, two ways to use computers to make progress in physics. The first approach is to use computers to compute a number, say the result of a certain integral or the hydrodynamic drag past a certain body. The second is to use computers as a kind of experimental laboratory, to explore the phenomena of interest much as an experimentalist would do him or herself. In the former case, realism is essential—if you do not solve the right equations, then you will not compute the right drag.