We now turn to the main topic of this book, disordered systems. We split this into two parts, treating in turn lattice models and mean-field models. From the physical point of view, the former should be more realistic and hence more relevant, so it is natural that we present the general mathematical framework in this context. However, the number of concrete problems one can to this day solve rigorously is quite limited, so that the examples we will treat can mostly be considered as random perturbations of the Ising model. In the second part we will be able to look, in a simplified setting, at more complex, genuinely random models, that rather surprisingly will turn out to produce fascinating mathematics, but also lead to applications that are beyond the standard scope of physics.

Introduction

We have seen that statistical mechanics is a theory that treats the dynamical degrees of freedom of a large system as random variables, distributed according to the Gibbs measure.

We now return to the class of Gaussian mean-field models whose covariance is a function of the Hamming distance, respectively the overlap RN. They were the original mean-field models for spin-glasses, introduced by D. Sherrington and S. Kirkpatrick in, and are considered to be the most natural, or physically relevant ones. Certainly, a large part of the excitement and interest, both in physics and in mathematics, has come from the fact that Parisi suggested a very complex solution, more precisely a formula for the free energy of these models, that has in many respects been seen as rather mysterious and unexpected. Moreover, this formula for the free energy suggested a structure of the Gibbs measures, which is to some extent similar to what we have encountered already in the CREMs. Much of the mystery of Parisis solution certainly resulted from the original method of its derivation, which involved objects such as zero-dimensional matrices, and that appeared to transcend the realm of standard mathematics.

Fortunately, these mysteries have been greatly clarified, mainly due to an ingenious discovery of F. Guerra in 2002, and subsequent work by Aizenman, Sims, and Starr, and M. Talagrand, that allows a clear and mathematically rigorous formulation of the Parisi solution.

The existence of the free energy

A key observation that has led to the recent breakthroughs in spin-glass theory, made by F.Guerra and F.-L. Toninelli, was that Lemma 10.2.1, or rather the interpolation idea used in its proof, is a powerful tool in this context.

Starting with the Newtonian revolution, the eighteenth and nineteenth centuries saw with the development of analytical mechanics an unprecedented tool for the analysis and prediction of natural phenomena. The power and precision of Hamiltonian perturbation theory allowed even the details of the motion observed in the solar system to be explained quantitatively. In practical terms, analytical mechanics made the construction of highly effective machines possible. Unsurprisingly, these successes led to the widespread belief that, ultimately, mechanics could explain the functioning of the entire universe. On the basis of this confidence, new areas of physics, outside the realm of the immediate applicability of Newtonian mechanics, became the target of the new science of theoretical (analytical) physics. One of the most important of these new fields was the theory of heat, or thermodynamics. One of the main principles of Newtonian mechanics was that of the conservation of energy. Now, such a principle could not hold entirely, due to the ubiquitous loss of energy through friction.

In the previous chapters we have seen that with considerable work it is possible to study some simple aspects of random perturbations of the Ising model. On the other hand, models that are genuinely random and promise to bring some new features to light are at the moment not seriously accessible to rigorous analysis. In such situations it it natural to turn to simplified models, and a natural reflex in statistical mechanics is to turn to the mean-field approximation. In the case of disordered systems, this has turned out, quite surprisingly, to open a Pandora's box.

The common feature of mean-field models is that the spatial structure of the lattice ℤd is abandoned in favour of a simpler setting, where sites are indexed by the natural numbers and all spins are supposed to interact with each other, irrespective of their distance. The prototype of all such models is the Curie–Weiss model, that we studied in Section 3.5.

We now turn to the question of what should be the natural class of disordered mean-field models to study? This question requires some thought, as there are at least two natural classes that propose themselves.

With the Hopfield model we have seen that the range of applications of statistical mechanics goes beyond standard physics into biology and neuroscience. In recent years, many such extensions have been noted. A particularly lively area is that of combinatorial optimization, notably with applications to computer science. While I cannot cover this subject adequately, I will in this last chapter concentrate on possibly the simplest example, the number partitioning problem. The connection to statistical mechanics has been pointed out by S. Mertens, from whom I learned about this interesting problem.

Number partitioning as a spin-glass problem

The number partitioning problem is a classical optimization problem: Given N numbers n1, n2, …, nN, find a way of distributing them into two groups such that their sums in each group are as similar as possible. One can easily imagine that this problem occurs all the time in real life, albeit with additional complication: Imagine you want to stuff two moving boxes with books of different weights. You clearly have an interest in making both boxes more or less the same weight, so that neither of them is too heavy. In computing, you want to distribute a certain number of jobs on, say, two processors, in such a way that all of your processes are executed in the fastest way, etcClearly, the two examples indicate that the problem has a natural generalization to partitionings into more than two groups. What is needed in practice is an algorithm that, when presented with the N numbers, finds the optimal partitioning as quickly as possible.

Statistical mechanics is the branch of physics that attempts to understand the laws of the behaviour of systems that are composed of very many individual components, such as gases, liquids, or crystalline solids. The statistical mechanics of disordered systems is a particularly difficult, but also particularly exciting, branch of the general subject, that is devoted to the same problem in situations when the interactions between these components are very irregular and inhomogeneous, and can only be described in terms of their statistical properties. From the mathematical point of view, statistical mechanics is, in the spirit of Dobrushin, a ‘branch of probability theory’, and the present book adopts this point of view, while trying not to neglect the fact that it is, after all, also a branch of physics.

This book grew out of lecture notes I compiled in 2001 for a Concentrated Advanced Course at the University of Copenhagen in the the MaPhySto programme and that appeared in the MaPhySto Lecture Notes series in the same year. In 2004 I taughta two-semester course on Statistical Mechanics at the Technical University of Berlin with in the curriculum of mathematical physics for advanced undergraduate students, both from thephysics and the mathematics departments. It occurred to me that the material I was going to cover in this course could indeed provide a suitable scope for a book, in particular as the mathematical understanding of the field was going through a period of stunning progress, and that an introductory textbook, written from a mathematical perspective, was maybe more sought after than ever.

We will now turn to the investigation of the rigorous probabilistic formalism of the statistical mechanics of lattice spin systems, or lattice gases. The literature on this subject is well developed and the interested student can find in-depth material for further reading in, and the classical monographs by Ruelle. A nice short introduction with a particular aim in viewis also given in the first sections of the paper.

Spin systems and Gibbs measures

As mentioned in the last chapter, the idea of the spin system was born in about 1920 in an attempt to understand the phenomenon of ferromagnetism. At that time it was understood that ferromagnetism should be due to the alignment of the elementary magnetic moments (‘spins’) of the (iron) atoms, that persists even after an external field is turned off. The phenomenon is temperature dependent: if one heats the material, the coherent alignment is lost. It was understood that the magnetic moments should exert an ‘attractive’ (‘ferromagnetic’) interaction towards each other, which, however, is of short range. The question was then, how such a short-range interaction could sustain the observed very long-range coherent behaviour of the material, and why such an effect should depend on the temperature.

Most computational methods in statistical mechanics rely upon perturbation theory around situations that are well understood. The simplest one is, as always, the ideal gas. Expansions around the ideal gas are known as high-temperature or weak-coupling expansions. The other type of expansions concern the situation when the Gibbs measure concentrates near a single ground-state configuration. Such expansions are known as low-temperature expansions. Technically, in both cases, they involve a reformulation of the model in terms of what is called a polymer model. We begin with the high-temperature case, which is both simpler and less model-dependent than the low-temperature case, and show how a polymer model is derived.

High-temperature expansions

We place ourselves in the context of regular interactions, and we assume that β will be small. In this situation, we can expect that our Gibbs measure should behave like a product measure. To analyze such a situation, we will always study the local specifications, establishing that they depend only weakly on boundary conditions.

About 1870, Ludwig Boltzmann proposed that the laws of thermodynamics should be derivable from mechanical first principles on the basis of the atomistic theory of matter. In this context, N moles of a gas in a container of volume V should be represented by a certain number of atoms, described as point particles (or possibly as slightly more complicated entities), moving under Newton's laws. Their interaction with the walls of the container is given by elastic reflection (or more complicated, partially idealized constraint-type forces), and would give rise to the observed pressure of the gas. In this picture, the thermal variables, temperature and entropy, should emerge as effective parameters describing the macroscopic essentials of the microscopic dynamics of the gas that would otherwise be disregarded.

The ideal gas in one dimension

To get an understanding of these ideas, it is best to consider a very simple example which can be analyzed in full detail, even if it is unrealistic.

The random-field Ising model has been one of the big success stories of mathematical physics and deserves an entire chapter. It will give occasion to learn about many of the more powerful techniques available for the analysis of random systems. The central question heatedly discussed in the 1980s in the physics community was whether the RFIM would showspontaneous magnetization at lowtemperatures and weak disorder in dimension three, or not. There were conflicting theoretical arguments, and even conflicting interpretations of experiments. Disordered systems, more than others, tend to elude common intuition. The problem was solved at the end of the decade in two rigorous papers by Bricmont and Kupiainen (who proved the existence of a phase transition in d ≥ 3 for small ∈) and Aizenman and Wehr (who showed the uniqueness of the Gibbs state in d = 2 for all temperatures).

The Imry–Ma argument

The earliest attempt to address the question of the phase transition in the RFIM goes back to Imry and Ma in 1975. They tried to extend the beautiful and simple Peierls argument to a situation with symmetry breaking randomness. Let us recall that the Peierls argument in its essence relies on the observation that deforming one ground-state, +1, in the interior of a contour γ to another ground-state, −1, costs a surface energy 2|γ|, while, by symmetry, the bulk energies of the two ground-states are the same. Since the number of contours of a given length L is only of order CL, the Boltzmann factors, e–2βL, suppress such deformations sufficiently to make their existence unlikely if β is large enough.