The first part of the book includes introductory concepts and background necessary for the understanding of anomalous diffusion in disordered media.

Fractals might be familiar to most readers, but their importance in modeling disordered and random media, as well as certain characteristics of the trails made by diffusing particles, makes it worthwhile to spend some time reviewing the subject. In Chapter 1 we provide working definitions of fractals, fractal dimensions, self-affine fractals, and related ideas. More importantly, we describe several algorithms for determining whether a particular object is a fractal, and for finding its fractal dimension. In the early days of fractal theory much effort was spent on merely exploring the fractal properties of various natural objects and physical models, using precisely such algorithms, and they continue to be essential tools for the study of disordered phenomena.

Percolation, which is reviewed in Chapter 2, is perhaps the most important model of disordered media and of naturally occurring fractals. Percolation owes its enormous appeal to its simplicity (it can be defined and analyzed using only geometrical concepts), its remarkably wide range of applications, and its being one of the most basic models of critical phase transitions. Relevant to our purpose is the fact that studies of anomalous diffusion have traditionally focused on percolation systems, and the problem still attracts considerable interest. The percolation transition, on the other hand, gives us an excellent opportunity to introduce several useful concepts, such as critical exponents, scaling, and the upper critical dimension.

In Chapter 3 we present a brief introduction to random-walk theory. Discrete random walks (in regular lattices) are discussed first, then diffusion and the diffusion equation are obtained as limiting cases.

The case of reversible coalescence, when the back reaction A → A + A is allowed, is special in that the steady state is a true equilibrium state. The process can then be analyzed by standard thermodynamics techniques and one expects simple, classical behavior. It is therefore surprising to find that the approach to equilibrium is characterized by a dynamical phase transition in the typical time of relaxation. This phase transition can be exactly analyzed in finite lattices, providing us with a unique opportunity to study finite-size effects in a dynamical phase transition.

The equilibrium steady state

We now turn to the case of reversible coalescence, when the back reaction A → A + A occurs at rate υ > 0 (and with no input, R = 0). The process is illustrated in Fig. 17.1. After a short transient the system arrives at a steady state with a finite concentration of particles. Because coalescence is now reversible, this steady state is in fact an equilibrium state, which satisfies detailed balance. The statistical time-reversible invariance of this equilibrium state can be seen in Fig. 17.1, in which the direction of time is ambiguous (compare it with Figs. 16.1 and 16.4).

The equilibrium state is a state of maximum entropy, and therefore the particles follow a completely random (Poisson) distribution. This is characterized by an exponential IPDF, peq(x) = ceqexp(−ceqx). Alternatively, the state may be described by the lack of correlation among the occupation probabilities of different sites: each site is occupied with probability ceq Δx, independent of other sites.

As discussed in Part III, diffusion–limited reactions are generally studied through a variety of approximation and computer–simulation techniques, since a comprehensive exact method of analysis has not yet been suggested. For this reason, exactly solvable models are of extreme importance: they serve as benchmark tests for the existing approximation and simulation methods; the exact techniques may hint at a more general approach, and serve as a basis for better approximations; and they contribute enormously to our understanding of the field as a whole.

In Part IV, we discuss a particular example of an exactly solvable model – that of diffusion-limited coalescence, A + A→ A, in one dimension. The model has been studied extensively by numerous researches, who have thought up an impressive amount of imaginative, elegant, solutions. The description of these works would require an additional volume, so they are acknowledged only in the bibliography. Instead, we limit ourselves to the method of interparticle distribution functions (IPDF), merely because we took part in its development and we understand it best. It should likewise be noted that neither is our model of choice the only one which can be solved exactly (very few others exist, though). The coalescence model yields an astonishingly wide range of kinetic behavior, well beyond what might be suspected from its stark simplicity. The restriction of the model to one dimension commonly draws criticism. This is dictated by the need to find exact solutions. On the other hand, recall that diffusion–limited kinetics is more anomalous the lower the dimension, so there is an advantage in studying one–dimensional models, in which differences from mean–field classical behavior are most pronounced.

The escape-rate formalism

One of the most interesting developments in the theory of irreversible processes is a connection, first explored by Gaspard and Nicolis, between the dynamical properties of open systems and the hydrodynamic or transport properties of such systems. We will explore this connection in Chapter 13, but first we must consider the dynamical properties of open systems. We consider a system to be open if the phase space of the system has physical boundaries and there is a mapping or transformation which can take phase points inside the boundaries to phase points outside the boundaries. Further, we will assume that the boundaries are such that once a phase point passes a boundary, it can never return to the bounded system. Thus the boundaries on the phase space region may be considered to be absorbing. To get some idea of the motivation for considering open systems, we might imagine a Brownian particle diffusing in a fluid inside a container with absorbing boundaries. The motion of the particle is really deterministic and can be described – microscopically – by some transformation in the phase space of the entire system. Now, when we describe the motion of the phase point, we lose the Brownian particle whenever it encounters the boundary of the container. If we were to describe the motion macroscopically, we would solve the diffusion equation for the probability density of the Brownian particle, in the fluid, with absorbing boundaries. The probability of finding the particle inside the container is an exponentially decreasing function of time with decay coefficient depending on the diffusion coefficient of the Brownian particle in the fluid, and on the geometry of the container.

We have just seen that in a system subjected to an external field and a thermostat which maintains the system's kinetic energy at a constant value there is a phase space contraction taking place. Prior to that, we showed that a fractal repeller can form in the phase-space for a Hamiltonian system with absorbing boundaries such that trajectories hitting the boundary never re-appear in the system. Some questions naturally arise as we think about these systems, such as:

1. To what kind of structure is the phase-space for a thermostatted system contracting?

2. What are the properties of such a structure? Is it a smooth hypersurface, say, or does it have a more complicated structure?

3. How do we describe fractal attractors and/or repellers and compute the properties of trajectories which are confined to them, since these objects are typically of zero Lebesgue measure in phasespace?

In this chapter, we will show that for each of these situations there is an appropriate measure that can be used to describe the resulting sets and to compute the properties of trajectories which are confined to these sets. Throughout our discussions we will suppose that the dynamics is hyperbolic. In the thermostatted case, the system contracts onto an attractor, and the attractor is characterized by an invariant measure which is smooth along unstable directions and fractal in the stable directions. Such measures are known as SRB (Sinai–Ruelle–Bowen) measures.

For a system with escape, the invariant measure on the repeller is different from that on an attractor, because escape takes place along the expanding directions.

Now we want to discuss a number of topics that are essential for an understanding of dynamical systems theory and which also play a role in a more detailed discussion of the relation between transport theory and dynamical systems theory. We begin with a discussion of the Kolmogorov-Sinai (KS) entropy, which is a characteristic property of those deterministic dynamical systems with ‘randomness’ properties similar to Bernoulli shifts discussed earlier. The KS entropy is essential for formulating the escape-rate expressions for transport coefficients, to be discussed in Chapters 11 and 12.

Heuristic considerations

Let us return for a moment to the Arnold cat map discussed in the previous chapter. There we illustrated an initial set, A, say, that is located in the lower left-hand corner of the unit square (see Fig. 8.3). As this set evolves under the action of the map TA, the set becomes longer and thinner so that after three iterations, the set has begun to fold back across the unit square, and after ten iterations, the set is so stretched out that it appears to cover the unit square uniformly (see Figs 8.5 and 8.6, respectively). Since the initial set A is getting stretched along the unstable direction, at every iteration we learn more about the initial location of the points within the initial set A That is, suppose that we can distinguish two points on the unit square only if they are separated by a distance δ, the resolution parameter, and suppose further that the characteristic dimension of the initial set, A, is of the order of δ.

Linear response theory

Linear response theory describes the changes that a small applied external field induces in the macroscopic properties of a system in equilibrium. The external field is supposed to be turned on at some initial time, when the system is in equilibrium, and then treated as a perturbation. As an example of linear response theory, we show how to use it to obtain the time–correlation-function expression – often called the Green–Kubo expression – for the electrical conductivity of a system that contains charged particles. If the applied electric field is small enough that heating effects can be ignored, then Ohm's law can be expressed as Je=σE, where Je is the electrical current density, E is the applied electric field, and σ is the electrical conductivity that we wish to compute. The time-correlation formula is an example of a set of formulae which relate transport coefficients in a fluid to time integrals of timecorrelation- functions. In the last section of this chapter, we will give an example of the derivation of such formulae for the case of tagged particle diffusion. First, we wish to examine one particular derivation of the formula for the electrical conductivity which has provoked a great deal of very instructive discussion, which, in turn, is closely connected to the general theme of this book.

We began this excursion into the dynamical systems approach to nonequilibrium statistical mechanics with a discussion of the Boltzmann transport equation. We end this excursion with the Boltzmann equation, but now we are going to use it to compute some Lyapunov exponents. The fact that the Boltzmann equation begins and ends this book may serve to illustrate both the power and the beauty of this equation, sitting at the heart of our understanding of irreversible phenomena.

The Lorentz gas as a billiard system

We are going to calculate the positive Lyapunov exponent for a two-dimensional hard-disk Lorentz gas. To do so, we will combine ideas of Boltzmann with those of Sinai, thus completing, in some sense, the transition from molecular chaos to dynamical chaos, and showing the deep connection between them. Imagine then a collection of hard disks of radius a placed at random in the plane at low density, i.e., na2 « 1, where n is the number density of the disks (see Fig. 18.1). Next, imagine a point particle moving with speed v in this array. The particle moves freely between collisions with the disks and makes specular collisions with the disks from time to time, preserving its speed and energy, but not its momentum upon collision. Sinai has considered some of the mathematical properties of this system, and has proved that it is mixing and ergodic. The moving particle has four degrees of freedom – two coordinates and two momenta – but the energy is conserved. Therefore, the phase-space of the moving particle is three-dimensional.