The object of the present chapter is small deviations from local equilibrium which are triggered by spontaneous fluctuations. In real fluids these fluctuations which temporarily disturb the system from local equilibrium are such that a fluid at global equilibrium can be viewed as a reservoir of excitations extending over a broad range of wavelengths and frequencies from the hydrodynamic scale down to the range of the intermolecular potential. Non-intrusive scattering techniques are used to probe these fluctuations at the molecular level (neutron scattering spectroscopy) and at the level of collective excitations (light scattering spectroscopy) (Boon and Yip, 1980). The quantity measured by these scattering methods is the power spectrum of density fluctuations, i.e. the dynamic structure factor S(k, ω) which is the space- and time-Fourier transform of the correlation function of the density fluctuations. The spectral function S(k, ω) is important because it provides insight into the dynamical behavior of spontaneous fluctuations (or forced fluctuations in non-equilibrium systems). Whereas the fluctuations extend continuously from the molecular level to the hydrodynamic scale, there are experimental and theoretical limitations to the ranges where they can be probed and computed. Indeed, no theory provides a fully explicit analytical description of space-time dynamics establishing the bridge between kinetic theory and hydrodynamic theory. Scattering techniques have limited ranges of wavelengths over which fluctuation correlations can be probed.

Since 1985, lattice gas automata have become a widely and actively explored field. Academic groups and industrial laboratories around the world have invested considerable effort in their research activities to drive the subject in various new and promising directions. As a result, the literature on lattice gases and related topics has grown so rapidly and has become so voluminous that an exhaustive list and a detailed review of all relevant lattice gas publications would practically make a book by itself.

Here we select research areas where lattice gases have played an important role, and, for each of them, we quote articles considered as representative, historically or presently. Our review is by no means complete and our choices are certainly selective; unavoidably we haven't done justice to those whose work escaped our selection. Nevertheless we have attempted to cover a range of works expanding over various aspects of the subject in the available literature. Our goal will be reached if the reader finds this chapter a helpful tool for exploring the subject beyond the scope of this book.

The historical ‘roots’

Discrete Kinetic theory

In the early sixties, the problem of shock waves in dilute gases was a subject of increasing research effort, for fundamental as well as industrial reasons.

As LGAs are constructed as model systems where point particles undergo displacements in discrete time steps and where configurational transitions on the lattice nodes represent collisional processes, one can view the lattice gas as a discretized version of a hard sphere gas on a regular lattice where particles are subject to an exclusion principle instead of an excluded volume. The advantage with LGAs is that, starting from exact microdynamical equations, statistical mechanical computations can be conducted rather straightforwardly in a logical fashion with well controlled assumptions to bypass the many-body problem. This is well exemplified by the development in Chapter 4 leading to the lattice Boltzmann equation. For the moment we shall consider the lattice gas automaton as a bona fide statistical mechanical model with extremely simplified dynamics. Nevertheless we may argue that the lattice gas exhibits two important features:

1. (i) it possesses a large number of degrees of freedom;

2. (ii) its Boolean microscopic nature combined with stochastic microdynamics results in intrinsic fluctuations.

Because of these spontaneous fluctuations and of its large number of degrees of freedom, the lattice gas can be considered as a ‘reservoir of thermal excitations’ in much the same way as a real fluid. Now the question must be raised – as for the hard sphere model in usual statistical mechanics – as to the validity of the lattice gas automaton to represent actual fluids. In Chapters 5 and 8 we consider full hydrodynamics and macroscopic phenomena.

The story of lattice gas automata started around 1985 when pioneering studies established theoretically and computationally the feasibility of simulating fluid dynamics via a microscopic approach based on a new paradigm: a fictitious oversimplified micro-world is constructed as an automaton universe based not on a realistic description of interacting particles (as in molecular dynamics), but merely on the laws of symmetry and of invariance of macroscopic physics. Imagine point-like particles residing on a regular lattice where they move from node to node and undergo collisions when their trajectories meet at the same node. The remarkable fact is that, if the collisions occur according to some simple logical rules and if the lattice has the proper symmetry, this automaton shows global behavior very similar to that of real fluids.

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.

It is one of the main objectives of statistical mechanics to provide a microscopic content to the phenomenologically established macroscopic properties and behavior of systems with many degrees of freedom. Although it is not necessary to have a complete knowledge of the details of the microscopic interactions to describe macroscopic phenomena in fluid systems, these phenomena emerge as a consequence of the basic dynamical processes. However, to establish rigorously the connection between the phenomenology and the underlying microscopic processes amounts to solving the many-body problem. Even for systems with oversimplified microscopic dynamics, such as lattice gas automata, this is an impossible task: approximations are unavoidable.

In this chapter we derive the equations governing the macroscopic dynamics of LGAs satisfying the semi-detailed balance condition; we shall start from the microscopic dynamics of the automaton, and use the lattice Boltzmann approximation (Suárez and Boon, 1997a,b). The main objective is to obtain the non-linear hydrodynamic equations, where the Euler and dissipative contributions are expressed in terms of the microscopic evolution rules of the automaton, and whose validity is not restricted to regions close to equilibrium, so that they can be used to analyze phenomena taking place in systems arbitrarily far from equilibrium, for instance in thermal LGAs under large temperature gradient.

In order to derive the hydrodynamic equations, we make use of the Boltzmann hypothesis (see Section 4.4.2) that particles entering a collision are uncorrelated.

One of our main objectives has been to show that single-species non-thermal lattice gases can exhibit large-scale collective behavior governed by the same continuous, isotropic and Galilean-invariant equations as real Newtonian fluids. This is true despite the intrinsically Boolean, spatially discrete, anisotropic and non-Galilean invariant structure of lattice gases. Moreover, in the past 10 years, further lattice gas models have been designed to incorporate more complicated physical features such as reactive processes, magneto-hydrodynamic phenomena or surface tension (see Section 11.4 in Chapter 11).

On one hand, there has been considerable effort in basic research to understand the subtleties of the statistical mechanics of lattice gases and on the other hand intense work has been accomplished to take advantage of the similarities between lattice gases and real fluids in order to simulate fluid motions with simple and easily implemented lattice gas algorithms. Indeed, because of their fully Boolean cellular automaton structure, lattice gases are excellent candidates for efficient implementations on both dedicated and general purpose computers with serial, vectorial, parallel or even massively parallel architecture. In addition, various physical effects can be added at low cost. For example, the presence in a flow of a rigid fixed obstacle is extremely easy to take into account: it just requires replacing the standard collision rule by a bounce-back rule (see Section 2.4.1) on all nodes covered by the obstacle. Modifying the shape or the position of the obstacle is almost immediate, and no mesh modification is necessary.

We now illustrate the abstract microdynamic notions of Chapter 2, with a presentation of lattice gas models in terms of the microdynamic tools. The models are chosen to illustrate the various microdynamical concepts; further models will be considered briefly in Chapter 11.

We start with the simplest two-dimensional model based on the square lattice, the earliest lattice gas model (1973) labeled HPP according to the initials of the authors: Hardy, de Pazzis and Pomeau. Sections 3.2 to 3.4 are devoted to models constructed on the triangular lattice and based on the FHP model initially introduced by Frisch, Hasslacher and Pomeau (1986). A ‘colored’ version of the FHP model, developed as a two-components lattice gas is presented in Section 3.5. A slightly more complex model, also based on the triangular lattice, but with thermal properties (Grosfils, Boon and Lallemand, 1992) is described in Section 3.6. We then move to three-dimensional systems in Section 3.7, as we introduce the basic (pseudo-four-dimensional) lattice gas model of d'Humières, Lallemand and Frisch (1986).

Except for the HPP model, all the models presented in this chapter, have been designed to exhibit large-scale dynamics in accordance with the Newtonian viscous behavior of isotropic fluids.

The HPP model

Historically, the first lattice gas model was introduced in the early seventies by Hardy, de Pazzis and Pomeau (1973) with motivations focusing on fundamental aspects of statistical physics (see also Hardy et al., 1972, 1976 and 1977).

Diffusion –limited reaction processes are those for which the transport time (the typical time until reactants meet) is much larger than the reaction time (the typical time until reactants react, when they are constrained to be within their reactionrange distance). The transport properties of the reactants largely determine the kinetics of diffusion –limited reactions. One then naturally wonders how the (often anomalous) diffusion of particles discussed so far may affect such processes. This, and the need to account for the effects of fluctuations in the concentration of the reactants at all length scales, as well as other sources of fluctuations, make the study of diffusion –limited reactions notoriously difficult. The topic is discussed in the next four chapters.

In Chapter 11, we begin with the far simple case of reaction –limited processes. In their case the system may be assumed to be homogeneous at all times: the transport mechanism and fluctuations play no significant role. The kinetics of reaction –limited processes is well understood, since they may be successfully analyzed by means of classical rate equations. We also touch upon the important subject of reaction –diffusion equations, but only at the mean –field level, without the addition of noise terms.

Chapter 12 discusses the Smoluchowski model for binary reactions, and trapping. It is instructive to see how diffusion –limited processes depart from their reaction –limited counterpart, even for such elementary reaction schemes.

This book is devoted to the special area of statistical mechanics that deals with the classical spin systems with quenched disorder. It is assumed to be of a pedagogical character, and it aims to help the reader to get into the subject starting from fundamentals. The book is supposed to be selfcontained (the reader is not required to go through all the references to understand something), being understandable for any student having basic knowledge of theoretical physics and statistical mechanics. Nevertheless, because this is only an introduction to the wide scope of statistical mechanics of disordered systems, in some cases to get to know more details about a particular topic the reader is advised to refer to the existing literature. Although throughout the book I have tried to present all the unavoidable calculations such that they would look as transparent as possible and have given everywhere (where it is at all possible) physical interpretations of what is going on, in many cases certain personal efforts and/or use of imagination are still required.

The first part of the book is devoted to the physics of spin-glass systems, where the quenched disorder is the dominant factor. The emphasis is made on a general qualitative description of the physical phenomena, being mostly based on the results obtained in the framework of the mean-field theory of spin-glasses with long-range interactions. First, the general problems of the spin-glass state are discussed at the qualitative level.

Nature abounds with types of structures for which loops may be neglected. The simplest example are perhaps linear polymers – modeled by self-avoiding walks – but also branched polymers (modeled by lattice animals), DLA aggregates, trees and tree-like structures, river systems, networks of blood vessels, and percolation clusters (in d ≥ 6) are common examples.

Diffusion in loopless structures is a lot simpler than that in other disordered substrates, for which loops cannot be neglected, and it therefore yields itself to a more rigorous analysis. Chiefly, anexact relation between dynamical exponents (the walk dimension and spectral dimension) and structural exponents (the fractal dimension and chemical length exponent) may be derived.

Diffusion in combs is a reasonable model for diffusion in some random substrates: the delay of a random walker caused by dangling ends and bottlenecks may be well mimicked by the time spent in the teeth of a comb. This case can be successfully analyzed with a CTRW and other techniques.

Loopless fractals

A large class of fractals are tree-like in structure. They are characterized by the absence of loops (or loops are so scarce that they may be neglected). In Figs. 7.1 and 7.2 we show examples of deterministic loopless fractals. For the study of transport properties it is useful to define their backbone, or skeleton. It consists of the union of all shortest (chemical) paths connecting the root of the tree with the peripheral sites.

The diffusion-limited coalescence model, A+AA, can be treated exactly in one dimension. The process is unexpectedly rich, displaying self-critical ordering in a nonequilibrium system, a kinetic phase transition, and a lattice version of Fisher waves. Thus, in spite of its simplicity it sheds light on many important aspects of anomalous kinetics. It also serves as a benchmark test for approximation methods and simulation algorithms. The coalescence model will concern us throughout the remainder of the book. Here we introduce the model and explain the technique which allows its exact analysis.

The one-species coalescence model

Our basic model is a lattice realization of the one-dimensional coalescence process A + A → A. The exact analysis can also be extended to the reversible process, A → A + A, as well as to the input of A particles. The system is defined on a one-dimensional lattice of lattice spacing Δx. Each site may be either occupied by an A particle or empty. The full process consists of the following dynamic rules.

Diffusion. Particles hop randomly to the nearest lattice site with a hopping rate 2D/(Δx)2. The hopping is symmetric, with rate D/(Δx)2 to the right and D/(Δx)2 to the left. At long times this yields normal diffusion, with diffusion coefficient D.

Birth. A particle gives birth to another at an adjacent site, at rate ν/Δx. This means a rate of ν/(2Δx) for birth on each side of the original particle. Notice that, while ν is a constant (with units of velocity), the rate ν/Δx diverges in the continuum limit of Δx → 0.