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.

Random fractals in Nature arise for a variety of reasons (dynamic chaotic processes, self-organized criticality, etc.) that are the focus of much current research. Percolation is one such chief mechanism. The importance of percolation lies in the fact that it models critical phase transitions of rich physical content, yet it may be formulated and understood in terms of very simple geometrical concepts. It is also an extremely versatile model, with applications to such diverse problems as supercooled water, galactic structures, fragmentation, porous materials, and earthquakes.

The percolation transition

Consider a square lattice on which each bond is present with probability p, or absent with probability 1 − p. When p is small there is a dilute population of bonds, and clusters of small numbers of connected bonds predominate. As p increases, the size of the clusters also increases. Eventually, for p large enough there emerges a cluster that spans the lattice from edge to edge (Fig. 2.1). If the lattice is infinite, the inception of the spanning cluster occurs sharply upon crossing a critical threshold of the bond concentration, p = pc.

The probability that a given bond belongs to the incipient infinite cluster, P∞, undergoes a phase transition: it is zero for p < pc, and increases continuously as p is made larger than the critical threshold pc (Fig. 2.2).

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.

This part of the book explores the subject of anomalous diffusion in fractals and disordered media. Our goal here is twofold: to introduce various approaches to the problem – exact, as well as approximate; and to become intimately acquainted with the phenomenon of anomalous diffusion, its characteristics, and its causes.

Chapter 5 discusses diffusion in the Sierpinski gasket. Anomalous diffusion is demonstrated through simulations and exact enumerations, and through an exact renormalization of the first-passage time. The important relation between diffusion and conductivity (the Einstein relation), introduced in Section 3.4, is used to rederive the anomalous diffusion exponent in yet another way. The chapter closes with a discussion of probability-density-distribution functions and of fractons and spectral dimensions.

In Chapter 6 we present a summary of diffusion in percolation clusters. The question of diffusion in the incipient infinite cluster versus diffusion in all of the clusters is analyzed through scaling and simulations. The chapter describes also the attempt of the Alexander–Orbach conjecture to connect between static and dynamic exponents, diffusion in chemical space, and the multifractality of conductivity and diffusion in percolation clusters.

Chapter 7 discusses diffusion in loopless fractals: Eden trees, combs, etc. In this important case some exact results may be derived, including general relations between static and dynamic exponents, which shed some light on the more general situation in which loops are relevant.

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).