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).

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.

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.

In this chapter the physical interpretation of the formal RSB solution will be proposed, and some new concepts and quantities will be introduced. The crucial concept that is needed to understand physics behind the RSB structures is that of the pure states.

The pure states

Consider again a simple example of the ferromagnetic system. Here, spontaneous symmetry breaking takes place below the critical temperature Tc, and at each site the non-zero spin magnetizations 〈σi〉 = ±m appear. As we have already discussed in Section 2.2, in the thermodynamic limit the two ground states with the global magnetizations 〈σi〉 = +m and 〈σi〉 = –m are separated by an infinite energy barrier. Therefore, once the system has happened to be in one of these states, it will never be able (during any finite time) to jump into the other one. In this sense, the observable state is not the Gibbs one (which is obtained by summing over all the states), but one of these two states with non-zero global magnetizations. To distinguish them from the Gibbs state they could be called the ‘pure states’. More formally, the pure states could also be defined by the property that all the connected correlation functions in these states, such as 〈σiσj〉c ≡ 〈σiσj〉 – 〈σi〉〈σj〉, tend towards zero at large distances.

In the previous chapter we obtained a special type of spin-glass ground-state solution.

Harris criterion

In studies of the phase-transition phenomena, the systems considered before were assumed to be perfectly homogeneous. In real physical systems, however, some defects or impurities are always present. Therefore, it is natural to consider what effect impurities might have on the phase-transition phenomena. As we have seen in the previous chapter, the thermodynamics of the second-order phase transition is dominated by large-scale fluctuations. The dominant scale, or the correlation length, Rc ∼ |T/TC – l|–v grows as T approaches the critical temperature Tc, where it becomes infinite. The large-scale fluctuations lead to singularities in the thermodynamical functions as |τ| ≡ |T/Tc – 1| → 0. These singularities are the main subject of the theory.

If the concentration of impurities is small, their effect on the critical behavior remains negligible so long as Rc is not too large, i.e. for T not too close to Tc. In this regime the critical behavior will be essentially the same as in the perfect system. However, as |τ| → 0 (T → Tc) and Rc becomes larger than the average distance between impurities, their influence can become crucial.

As Tc is approached the following change of length scale takes place. First, the correlation length of the fluctuations becomes much larger than the lattice spacing, and the system ‘forgets’ about the lattice. The only relevant scale that remains in the system in this regime is the correlation length Rc(τ).