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.

Until now we have considered systems involving noninteracting walkers. This is an enormous simplification that allows analysis of such problems in terms of a single walker. Reality, however, is more complex and interactions cannot always be neglected. In this chapter we consider an elementary type of hard-core repulsion, known as excluded-volume interactions: walkers, or particles, are not allowed to occupy the same site simultaneously. We describe the dramatic effects that this simple interaction has on diffusion.

A self-avoiding walk (SAW) is a random walk that does not intersect itself. SAWs are a useful model for linear polymer chains: the visited sites represent monomers, and self-avoidance accounts for the excluded-volume interactions between monomers. The study of polymers in random media finds applications in enhancing recovery of oil, gel electrophoresis, gel-permeation chromatography, etc. Flory's theory provides us with a beautiful, intuitive understanding of the anomalous properties of SAWs in regular Euclidean space, and it may be extended to percolation and fractals. The problem of SAWs in finitely ramified fractals can be solved exactly.

Tracer diffusion

Imagine a regular lattice of lattice spacing a with a density c of particles, i.e., each site is occupied with probability c. The particles perform nearest-neighbor random walks, with hopping rate Г, and are subject to excluded-volume interactions: at most one particle may occupy a site at any given moment. Clearly, diffusion of the particles is hindered by these interactions. For example, in the limit c = 1, when all sites are occupied, motion is impossible and the system is frozen.

In this chapter we will consider classical experiments that have been performed on real spin glass materials, aiming to check to what extent the qualitative picture of the spin-glass state described in previous chapters does take place in the real world. The main problem of the experimental observations is that the concepts and quantities that are very convenient in theoretical considerations are rather far from the experimental realities, and it is a matter of the experimental art to invent convincing experimental procedures that would be able to confirm (or reject) the theoretical predictions.

A series of such brilliant experiments has been performed by M. Ocio, J. Hammann, F. Lefloch and E. Vincent (Saclay), and M. Lederman and R. Orbach (UCLA) [9]. Most of these experiments have been done on the crystals CdCr1.7In0.3S4. The magnetic disorder there is present due to the competition of the ferromagnetic nearest neighbor interactions and the antiferromagnetic higher-order neighbor interactions. This magnet has already been systematically studied some time ago [26], and its spin-glass phase transition point T = 16.7 K is well established. Some of the measurements have been also performed on the metallic spin glasses AgMn [27] and the results obtained were qualitatively quite similar. It indicates that presumably the qualitative physical phenomena observed do not depend very much on the concrete realization of the spin-glass system.

In this chapter we present a new method for studying statistical systems with quenched disorder in the low-temperature limit. The use of the replica method has turned out to be very efficient in some disordered systems. It allows for a detailed characterization of the low-temperature phase at least at the mean-field level. In all the mean-field spin-glass-like problems where one can expect the mean-field theory to be exact, the Parisi scheme of replica symmetry breaking is successful, and at the moment there is no counterexample showing that it does not work. On the other hand, the low-temperature phase of these systems is complicated enough, even at the mean-field level. One might hope that the very low-temperature limit could be easier to analyse, while its physical content should be basically the same. This very low-temperature limit is also an extreme case where one might hope to get a better understanding of the finite-dimensional problems. At first sight the low-temperature limit is indeed simpler because the partition function could be analysed at the level of a saddle-point approximation. However, it is easy to see that generically this limit does not commute with the limit of the number of replicas going to zero. There is a very basic origin to this non-commutation, namely the fact that there still exist, even at zero temperature, sample-to-sample fluctuations.