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.

Although the microscopic makeup of fluids ranges from the simplest monatomic gas to, say, a complex mixture such as milk, nearly all fluids flow in a way that obeys the same equations of fluid mechanics. How simple can a model of a fluid be and still satisfy these same equations?

In this chapter we introduce a microscopic model of a fluid that is far simpler than any natural fluid. Indeed, it has nearly nothing in common with real fluids except for one special property—at a macroscopic scale it flows just like them!

This simple model represents an attempt to digitize, or reduce to logic, the equations of motion of hydrodynamics. After a discussion of the model's historical relation to other such attempts to simplify physics to make it more adaptable to computation, we consider some of the specific ramifications of the discovery of this simple fluid. This chapter establishes the context for the more detailed analyses, extensions, and applications of this model that follow.

The lattice gas

In 1986, Uriel Frisch, Brosl Hasslacher, and Yves Pomeau announced a striking discovery. They showed that the molecular, or atomistic, motion within fluids—an extraordinarily complicated affair involving on the order of 1024 real-valued degrees of freedom—need not be nearly so detailed as real molecular dynamics in order to give rise to realistic fluid mechanics. Instead, a fluid may be constructed from fictitious particles, each with the same mass and moving with the same speed, and differing only in their velocities.

In the previous chapters we have restricted our discussion to models of simple fluids such as water. But what if the water contains some dye? Thus we now consider models of fluid mixtures.

In this chapter we consider the simplest mixtures—those composed of two miscible fluids. We will construct models of miscible fluids from straightforward extensions of the models of the previous chapters. To illustrate an application of these miscible-fluid models, we will close this chapter with a summary of a study of passive scalar dispersion in a slow flow.

A discussion of miscible lattice gas mixtures is the first step toward an understanding of lattice gas models of immiscible fluids, a subject which we shall take up in the following chapter.

Boolean microdynamics

Miscible lattice-gas mixtures are constructed by adding a second type of particle and letting it evolve passively. This is roughly equivalent to injecting a fluid with a colored dye that allows one to see the fluid motions but does not affect the flow itself.

For convenience, we usually distinguish between particle type by assuming that the particles are colored. Our favorite mixtures are red and blue. (Aside from a certain aesthetic appeal, this choice also nicely avoids the political pitfalls of black and white.) In a miscible lattice-gas mixture the color of the particles is a property that is carried with them, but the evolution of the particles is no different than it would be if they were not colored.

One of the greatest virtues of lattice-gas models and lattice-Boltzmann methods is the ease with which they allow one to include microscopic complexity in a model of a fluid. Throughout much of this book we have already considered models of immiscible two-fluid mixtures, the collision rules of which include interactions between particles at neighboring sites. In this chapter, we consider some of the ways in which these ideas may be generalized to create fluids of even greater complexity.

We will cover a fair bit of ground. The models range from toys for the study of pattern formation to methods for the simulation of multi-phase flow. Aside from their intrinsic interest as models of complex fluids, these models also serve to indicate how collision rules may be designed to introduce other kinds of microscopic physics into momentum-conserving hydrodynamic models.

We proceed roughly in the order of increasing complexity.

Stripes and bubbles

We first describe a model that produces some fascinating patterns. Although it conserves momentum, our discussion here is motivated less by fluid mechanics than by pattern formation itself.

The model is a nearly trivial extension of the 2D immiscible lattice gas (ILG) of Chapter 9. In the ILG, red particles and blue particles interact in a way that results in a kind of short-range attraction between particles of the same color. We now introduce a competing long-range repulsion.

We employ the same notation as we used in Section 9.1.

Whereas phase separation is undoubtedly the most striking aspect of immiscible lattice-gas models, the interfaces that form due to phase separation are themselves an object of at least equal interest. In this chapter, we consider the interfaces formed by the 2D immiscible lattice-gas (ILG) model of Chapter 9.

We first discuss a theoretical calculation of the surface tension. Our calculation of surface tension not only provides a better understanding of ILG interfaces, but it also predicts the phase transition from the mixed to the unmixed phase described earlier in Chapter 16.

We then present a detailed view of interface fluctuations. In real fluids, interfaces fluctuate due to thermal noise. Lattice-gas interfaces, on the other hand, fluctuate due to the statistical noise in the Boolean dynamics. In both cases, the detailed motion of the interfaces results from a combination of surface tension, viscous hydrodynamics, and noisy excitation. We shall see that a study of interface fluctuations provides a delicate probe of the hydrodynamic and statistical properties of the ILG.

Surface tension: a Boltzmann approximation

The calculation of surface tension in ILG's offers neither the elegance nor the accuracy of the analogous calculation for lattice-Boltzmann models that we presented in Chapter 10. It does however yield several interesting results.

As in Section 10.2, we once again consider the surface tension of a flat interface. As we have already indicated in Exercise 11.3, there is no reason to expect that surface tension is isotropic in Boolean models.

Phase separation—the spontaneous separation of two initially mixed fluids—is one of the most fascinating aspects of immiscible lattice-gas mixtures. As we have already seen in Chapters 9, 11, and 12, phase separation occurs as a result of a phase transition in lattice-gas models. Though the resulting phase-separation dynamics can be quite dramatic, the transition itself can be difficult to describe. In particular, non-equilibrium aspects of lattice-gas phase transitions remain to be fully addressed.

Using a combination of theoretical arguments and numerical simulation, we focus in this chapter on the characterization of phase separation in lattice gases. We begin with a brief review of a classical model of phase separation. Then, focusing on the specific case of the immiscible lattice-gas (ILG) models of Chapters 9 and 11, we detail some of the ways in which our discrete models have been shown thus far to qualitatively (and sometimes quantitatively) reproduce the non-equilibrium evolution of real phase separation.

Phase separation in the real world

Phase separation occurs when the mixed state of a mixture is unstable, so that its components spontaneously segregate into bulk phases composed primarily of one species or the other. If the instability results from a finite, localized perturbation of concentration in the mixture, it is known as nucleation. If instead the perturbation is infinitesimal in amplitude, not localized, and of sufficiently long wavelength, the instability is known as spinodal decomposition.

We now take a break from the theoretical developments of the previous chapters and consider applications of lattice gases to the study of complex flows through complex geometries. Our complex geometry is one of the most complicated nature has to offer—a porous rock. The flows we consider are either those of a simple fluid, such as water, or an immiscible two-fluid mixture, such as water and oil. The problems we shall illustrate are not only of intrinsic interest for physics but have applications in fields as diverse as hydrology, oil recovery, and biology, to name just a few.

Our objectives in this chapter are twofold. First, we wish to indicate the level of accuracy that one may expect from these kinds of flow simulations. Second, we wish to show what we can learn from such work. We begin with a brief introduction to the subject.

Geometric complexity

All rocks found on the earth's surface are porous, but those rocks that we call sandstones are usually more porous than others. Sandstones are formed from random assemblages of sand grains that are cemented together over geologic time. Fluids such as oil or water may then become trapped in the pore space between the cemented sand grains. There may be an economic interest in extracting these fluids, or, equally possible in modern times, we may wish to predict the rate at which some contaminant such as radioactive waste could migrate through such a medium.