Today, there is a growing interest in understanding the role of chaos in nonequilibrium statistical mechanics. Although ergodic theory has been one of the seeds of modern dynamical systems theory, it is only recently that new methods have been developed – especially, in periodic-orbit theory – in order to quantitatively characterize the microscopic chaos as well as the intrinsic rates of decay or relaxation of statistical ensembles of trajectories. One of these intrinsic rates is the escape rate associated with the so-called fractal repeller which plays a central role in chaotic scattering. During recent years, chaotic scattering has been discovered in many different fields, from celestial mechanics and hydrodynamics to atomic, molecular, mesoscopic, and nuclear physics. In molecular systems, chaotic scattering provides a classical and statistical understanding of chemical reactions. Chaotic scattering is also closely related to transport processes like diffusion or viscosity. In this way, relationships can be established between the transport coefficients and the characteristic quantities of microscopic chaos, such as the Lyapunov exponents, the Kolmogorov–Sinai entropy, or the fractal dimensions. These results and their developments shed new light on nonequilibrium statistical mechanics and the problem of irreversibility.

The aim of the present book is to describe the theory of chaotic scattering and this new approach to nonequilibrium statistical mechanics starting from the principles of dynamical systems theory and from the hypothesis of microscopic chaos. For lack of space and time, the book only contains results on classical dynamical systems, although many fascinating and closely connected results have also been obtained in the context of quantum dynamics.

Dynamical randomness and the entropy per unit time

If dynamical instability is quantitatively measured by the Lyapunov exponents, on the other hand, dynamical randomness is characterized by the entropy per unit time. The entropy per unit time is a transposition of the concept of thermodynamic entropy per unit volume from space translations to time translations. As Boltzmann showed, the entropy is the logarithm of the number of complexions, i.e., the number of microscopic states which are possible in a certain volume and under certain constraints. In the time domain, the number of complexions becomes the number of possible trajectories in a given time interval. The entropy per unit time is therefore an estimation of the rate at which the number of possible trajectories grows with the length of the time interval.

This scheme is not in contradiction with the famous Cauchy theorem which asserts the uniqueness of the trajectory issued from given initial conditions. Indeed, as in statistical mechanics, the counting proceeds with the constraint that the trajectories belong to cells of phase space. Since each cell is a continuum, the counting becomes nontrivial. Indeed, an initial cell may be stretched into a long and thin cell which will overlap several other cells at the next time step. In this way, the stretching and folding mechanism in phase space implies that the tree of possible trajectories has a number of branches which grows exponentially with a positive branching rate.

The counting may be purely topological, which yields the definition of the topological entropy per unit time of Chapter 2.

The idea that gases are disordered or amorphous states of matter is old. Actually, the word gas was created from the Greek word chaos by Joan-Baptista van Helmont (1577–1644). This Flemish physician and chemist born in Brussels was the first to distinguish different kinds of gases thanks to the experimental method and he also invented an air thermoscope which was the precursor of the modern thermometer. He was contemporary with Bacon (1561–1626), Galileo (1564–1642), Kepler (1571–1630), Descartes (1596–1650), Torricelli (1608–1647), as well as with the famous painter Rubens (1577–1640). His son published his work Ortus medicinae, id est initia phisicare inaudita at Amsterdam in 1648 (Farber 1961).

During the XlXth century, the spatial disorder of gases and of matter in general was quantitatively characterized with the concept of entropy per unit volume. However, the idea of dynamical chaos, i.e., of temporal disorder in physical systems like gases is more recent as it results from a long sequence of observations and works which extends throughout the XXth century with the development of statistical mechanics.

Today, we may say that statistical mechanics and kinetic theory are among the greatest successes of modern science. Since Maxwell and Boltzmann, macroscopic properties of matter can be explained in terms of the motion of atoms and molecules composing matter. In particular, transport properties like diffusion, viscosity, or heat conductivity can be predicted in terms of the parameters of the microscopic Hamiltonians, which are the masses of the atoms and molecules, and the coupling constants of their interaction (Maxwell 1890, Boltzmann 1896).

Hydrodynamics from Liouvillian dynamics

Historical background and motivation

Hydrodynamics describes the macroscopic dynamics of fluids in terms of Navier–Stokes equations, the diffusion equation, and other phenomenological equations for the mass density, the fluid velocity and temperature, or for chemical concentrations. In nonequilibrium statistical mechanics, these phenomenological equations may be derived from a kinetic equation like the famous Boltzmann equation or other master equations describing the time evolution at the level of one-body distribution functions (Balescu 1975, Résibois and De Leener 1977, Boon and Yip 1980). The kinetic equation itself is derived from Liouvillian dynamics using a Markovian approximation such as Boltzmann's Stosszahlansatz. Such approximations may be justified in some scaling limits for dilute fluids or other systems, but the derivation of hydrodynamics is not carried out directly from the Liouvillian dynamics. The only direct link between hydrodynamics and the Liouvillian dynamics – which is used in particular in molecular-dynamics simulations – is established in terms of the Green–Kubo relations.

The recent works in dynamical systems theory have shown that further direct links are possible. In particular, we have observed with the multibaker map in Chapter 6 that the spectrum of the Pollicott–Ruelle resonances actually provides the spectrum of the phenomenological diffusion equation in spatially extended systems (Gaspard 1992a, 1995, 1996). This result suggests that the dispersion relations of hydrodynamics can be obtained in terms of the Pollicott– Ruelle resonances and that the hydrodynamic modes can be constructed as the associated eigenstates.

Classical scattering theory

Motivations

Matter is often studied by scattering with beams of particles such as photons, electrons, neutrons, or others. The quantities of interest are the cross-sections which give the effective surface offered by the target for the realization of a certain scattering event. Scattering processes are usually conceived in a statistical approach. For instance, a cross-section cannot be determined by a single collision but by a statistical ensemble of collisions with a uniform distribution of the incoming impact parameters. In this regard, a natural relation appears between scattering theory and the Liouvillian dynamics.

Many different processes may be considered in scattering theory, for instance elastic or inelastic collisions (Joachain 1975). Among the latter, the reaction processes between molecules or nuclei are of particular importance because they play a crucial role in the transformation of matter. Beside the cross-sections, other important quantities are the reaction rates which characterize the time evolution of statistical ensembles during reactions. The rates have the inverse of a time as unit. We may thus expect that reaction rates belong to the same class of properties as the relaxation rates of Liouvillian dynamics. This is the case, in particular, for unimolecular reactions which are dissociation processes (Gaspard and Rice 1989a, 1989b). The reaction rates can here be assimilated with the lifetimes of the metastable states of the transition complex, i.e., of the transient states formed when the fragments of the reactions are still in interaction. Here also, these lifetimes are essentially statistical properties of the time evolution instead of properties of individual trajectories of the system.

Nonequilibrium systems in Liouvillian dynamics

Most systems in nature are maintained out of equilibrium either by incident fluxes of particles or by external fields. The earth bathed by sunlight1 is an illustration of such out-of-equilibrium systems. From this viewpoint, the systems may be considered as subjected to some scattering processes, which leads us to the scattering theory of transport of Chapter 6. In this context, the fact that most classical scattering processes are chaotic has important consequences in our understanding of nonequilibrium states and the methods of the previous chapters are thus required for the investigation of out-of-equilibrium systems.

Works on out-of-equilibrium systems have revealed that such systems remain in a thermodynamic state which is the continuation of the equilibrium state under weak nonequilibrium constraints. Beyond a certain threshold, the thermodynamic branch becomes unstable and new states emerge by bifurcation with spatial or temporal inhomogeneities, called dissipative structures (Prigogine 1961; Glansdorff and Prigogine 1971; Nicolis and Prigogine 1977, 1989). Turing structures in reaction–diffusion systems and convection rolls in fluids are examples of such nonequilibrium structures (DeWit et al. 1992, 1993, 1996; Cross and Hohenberg 1993). The transitions to dissipative structures appear sharp from a macroscopic viewpoint which ignores the thermodynamic fluctuations due to the atomic structure of matter. These fluctuations can be modelled by stochastic dynamical systems like Langevin processes, birth-and-death processes, or lattice-gas automata, which show that transitions may be rounded in systems with finitely many particles (Nicolis and Malek Mansour 1978, Malek Mansour et al. 1981, Dab et al. 1991, Lawniczak et al. 1991, Kapral et al. 1992, Baras and Malek Mansour 1997).

Ideas from the theory of critical phenomena have been of great importance in the modelling of polymers ever since the Nobel prize winner P. G. de Gennes showed (in 1972) how the two subjects can be connected. In the 25 years that have passed since then, almost every major development in the understanding of criticality has led to parallel progress in the study of polymer models. We can think of the renormalisation group, the introduction of ideas from fractal geometry, conformal invariance …. As a result of all this work, the equilibrium behaviour of a polymer in a diluted regime is by now very well understood. That's why I considered the time ripe to write an overview of this field of research.

There already exist excellent books on the statistical mechanics of polymers and it may therefore be important to say a few words about the ‘niche’ in which this book should be placed. I have put the emphasis on models on a lattice and have therefore said very little about models, and methods to treat them, which work in the continuum. For completeness, important results obtained in the continuum are mentioned, but it would take much more space (and expertise on the part of the author) to treat them in all detail. Moreover, they have been very well described, for example in. This book also deals almost exclusively with the very dilute regime in which we can study one isolated polymer and can neglect the influence of any other polymers which may be present.

In the previous three chapters we considered the behaviour of polymers in bulk. In reality systems are never infinite and one always has to consider the presence of surfaces. When the polymer is close to or even attached to a surface its critical properties may change. When there is an interaction between the monomers and the surface interesting adsorption effects can occur. We now turn to a discussion of these phenomena.

Surface magnetism

Consider a (d > 1)-dimensional lattice, in which a polymer is restricted to be in a semi-infinite region, e.g. the region with x ≥ 0. We imagine a wall at x = 0 which is impenetrable. When the polymer is very far from this surface, i.e. in the bulk, its properties are hardly changed by the presence of the wall. As the polymer is placed closer and closer to the surface its properties may be modified. These effects can be expected to have a scaling behaviour depending on the ratio of the distance xcm of the centre of mass of the polymer from the surface and its radius RN. As we are interested in properties of the polymer on a coarse grained scale, let us immediately attach one of the monomers to the surface (figure 5.1). There are two ways to discuss the properties of such a polymer. The first one works directly with SAWs, while the other one uses, through the O(n)-connection, known properties of surface magnetism. Since we don't expect the general reader to know about surface critical behaviour, we will give a brief overview of the main ideas from this field (for a general review, see).

So far we have studied SAWs in the low fugacity regime where z ≤ μ-1. When z > μ-1, the grand partition function diverges. We can still give a meaning to it by making an analytic continuation. On the other hand, we know that SAWs appear in the high temperature expansion of the O(n)-model and that μ-l corresponds to the critical temperature of that model. The regime where z > μ-1 therefore corresponds to the low temperature phase of the O(n)-model. A spin model has of course a well defined low temperature regime and one might ask what this phase means for polymers. It is these questions which we study in the present chapter. The polymers in this phase are usually referred to as dense polymers. Our discussion will mostly be limited to the two-dimensional case.

The low temperature region of the O(n)-model

The study of polymers in the high fugacity regime can be performed in essentially two ways. The first way is to study the low temperature properties of the O(n)-model. This will be done using the techniques known from previous chapters; the Coulomb gas, exact solutions using the Bethe Ansatz, and so on. On the other hand we can study immediately the properties of walks themselves, in the regime where z > μ-1. Sure enough, in that region the grand partition function diverges, but the trick is to study the properties of walks in finite systems, e.g. in a finite box of volume Λ. A typical size for such a volume is Λ1/d. The finite volume leads to a cutoff for the grand partition sum.

In this chapter we make a side step to a subject which is not directly concerned with polymers but which is closely connected to it. It is the study of percolation. Together with problems such as SAWs and lattice animals percolation forms a subject which is sometimes called geometrical critical phenomena. As we will see in chapter 8, percolation (in d = 2) is closely related to the behaviour of polymers at the ‘θ-point’ Moreover, percolation plays a role in the collapse of branched polymers. We will also discuss how percolation is related to a spin model (the Potts model) just as polymers are related to the O(n)-model. This Potts model turns out to be of importance in the description of dense polymer systems (chapter 7). The Potts model can also be used to describe the so called spanning trees. These are in turn interesting in the study of branched polymers (chapter 9).

In this book we have to limit ourselves to a discussion of those properties of percolation and Potts models which are necessary for the sequel of this book. There exists excellent reviews and books about percolation and for more information we refer to these.

Percolation as a critical phenomenon

To introduce percolation, think of a regular lattice, e.g. the hypercubic lattice in d dimensions. Take a real number 0 ≤ p ≤ 1 which we call the occupation probability. We occupy either the vertices (sites) or the edges (bonds) of this lattice with probability p.