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.

Introduction

Statistical mechanics is a very fruitful and successful combination of (i) the basic laws of microscopic dynamics for a system of particles with (ii) the laws of large numbers. This branch of theoretical physics attempts to describe the macroscopic properties of a large system of particles, such as one would find in a fluid or solid, in terms of the average properties of a large ensemble of mechanically identical systems which satisfy the same macroscopic constraints as the particular system of interest. The macroscopic phenomena that concern us in this book are those which fall under the general heading of irreversible thermodynamics, in general, or of fluid dynamics in particular. We shall be concerned with the second law of thermodynamics, more specifically, with the increase of entropy in irreversible processes. The fundamental problem is to reconcile the apparent irreversible behavior of macroscopic systems with the reversible, microscopic laws of mechanics which underly this macroscopic behavior. This problem hats actively engaged physicists and mathematicians for well over a century.

The law of large numbers and the laws of mechanics

Many features of the solution to this problem were clear already to the founders of the subject, Maxwell, Boltzmann, and Gibbs, among others. The notion that equilibrium thermodynamics and fluid dynamics have a molecular basis is one of the central scientific advances of the 19th century. Of particular interest to us here is the work of Maxwell and Boltzmann, who tried to understand the laws of entropy increase in spontaneous natural processes on the basis of the classical dynamics of many-particle systems.

We can now assemble many but, as we shall see, not all, of the pieces we need to construct a consistent picture of the dynamical foundations of the Boltzmann equation and similar stochastic equations used to describe the approach to equilibrium of a fluid or other thermodynamic system. While there are many fundamental points which still are in need of clarification and understanding, our study of hyperbolic systems with few degrees of freedom has pointed us in some interesting directions. In earlier chapters, we saw that the baker's transformation is ergodic and mixing. Moreover, when one defines a distribution function in the unstable direction, one obtains a ‘Boltzmann-like’ equation with an Htheorem. That is, there exists an entropy function which changes monotonically in time until the distribution function reaches its equilibrium value, provided the initial distribution is sufficiently well behaved, e.g., not concentrated on periodic points of the system. Moreover, the approach to equilibrium takes place on a timescale which is determined by the positive Lyapunov exponent and is typically shorter than the time needed for the full phase-space distribution function function to be uniformly distributed over the phase-space. Although we can make all of this clear for the baker's transformation it is not so easy to reproduce these arguments in any detail for realistic systems of physical interest. However, we can study more complicated hyperbolic maps to isolate the features we expect to use in a deeper discussion of the Boltzmann equation itself.

We have now arrived at a point where we can begin to see what all of the discussions in the previous chapters are leading to. That is, we can now make connections between the dynamical and transport properties of Anosov systems. In this chapter, we discuss two new approaches to the statistical mechanics of irreversible processes in fluids that use almost all of the ideas that we have discussed so far. These are the escape-rate formalism of Gaspard and Nicolis, and the Gaussian thermostat method due to Nose, Hoover, Evans and Morriss. It should be mentioned at the outset that this is a new area of research, that many more developments can be expected from this approach to transport, and that what we will discuss here are merely the first glimmerings of the results that can be obtained by thinking of transport phenomena in terms of the chaotic properties of reversible dynamical systems. There is a third, closely related, dynamical approach to transport coefficients based upon the properties of unstable periodic orbits of a hyperbolic system. We will discuss this approach in Chapter 15.

The escape-rate formalism

Suppose we think of a system that consists of a particle of mass m and energy E, moving among a fixed set of scatterers which are in some region R which is of infinite extent in all directions except one, the x-direction, such that the scatterers are confined to the interval 0 ≤ x ≤ L. Absorbing walls are placed at the (hyper) planes at x = 0 and x = L (see Fig. 12.1).

In this chapter, we will discuss briefly some simple models of fluid systems that are designed to exhibit many of the nonequilibrium properties of a real fluid, and to be very suitable for precise computer studies of fluid flows since only binary arithmetic is used to simulate these models. The models were devised by Prisch, Hasslacher, and Pomeau, among others, and are generally called cellular automata lattice gases. The corresponding one-dimensional Lorentz gas, studied in great detail by Ernst and co-workers, may be viewed as a ‘modern-day’ Kac ring model. The interest of these models for us consists in the fact that it is rather straightforward to compute both the transport as well as the chaotic properties of these systems, and the thermodynamic formalism is especially useful here. After introducing the general class of cellular automata lattice gases (CALGs) we will turn our attention to the special case of the one-dimensional Lorentz lattice gas (LLG) to outline how its dynamical quantities can be calculated.

Cellular automata lattice gases

Consider a two-dimensional hexagonal or square lattice with bonds connecting the nearest-neighbor lattice sites. A CALG is constructed by (i) putting indistinguishable particles on this lattice with velocities that are aligned along the bond directions, (ii) considering that the time is discretized, and (iii) stating that in one time step a particle goes from one site to the next in the direction of its velocity. The number of possible velocities for any particle is then equal to the coordination number, b, of the lattice, although models with rest particles (zero velocity), or with other velocities, are often considered.

In the course of our discussions of the baker's map, we noticed that we could easily use its isomorphism with the Bernoulli sequences to locate periodic orbits of the map. As we show below, we can exploit this isomorphism to prove that periodic orbits of the baker's map form a dense set in the unit square. Moreover, we will prove, without much difficulty, that the periodic orbits of the hyperbolic toral automorphisms are also dense in the unit square (or torus). A natural question to ask is: If these periodic orbits are ubiquitous, can they be put to some good use? In this chapter, we outline some simple affirmative answers to this question in the context of nonequilibrium statistical mechanics. In particular, we will see that periodic orbit expansions are natural objects when one encounters the need for the trace of a Frobenius–Perron operator, and when one wants to make explicit use of an (∈, T)-separated set. Moreover, the periodic orbits of a classical system form a natural starting point for a semi-classical version of quantum chaos theory. We should also mention that there is a new field of study dealing with issues related to the control of chaos, which exploits the presence of periodic orbits to slightly perturb a system from chaotic behavior to a more easily controlled periodic behavior.

Dense sets of unstable periodic orbits

Here we consider a hyperbolic system. If we have located a periodic orbit of our system, then each point on it has a set of stable and unstable directions.

The transformation and its properties

We return to our discussion of dynamical systems, and consider an example of great illustrative value for the applications of chaos theory to statistical mechanics, the baker's transformation. For this example, we take the phase-space to be a unit square in the (x,y)-plane, with 0 ≤ x,y≤ 1. The measure-preserving transformation will be an expansion in the x-direction by a factor of 2 and a contraction in the y-direction by a factor of 1/2, arranged in such a way that the unit square is mapped onto itself at each unit of time.

The transformation consists of two steps (see Fig. 7.1): First, the unit square is contracted in the y-direction and stretched in the y-direction by a factor of 2. This doesn't change the volume of any initial region. The unit square becomes a rectangle occupying the region 0≤x ≤ 2; 0 ≤ y ≤ 1/2. Next, the rectangle is cut in the middle and the right half is put on top of the left half to recover a square. This doesn't change volume either. This transformation is reversible except on the lines where the area was cut in two and glued back.

We have now covered the background material needed to approach the literature on dynamical systems theory and nonequilibrium statistical mechanics. Here we list a few topics that you might find stimulating to think about. Some references are provided, but you should spend some time on the computer looking up relevant papers in areas that you find especially interesting.

Very nice overviews of this field of research are provided by D. Ruelle and Ya. G. Sinai in their paper ‘Prom dynamical systems to statistical mechanics and back’ [RS86]; in Ruelle's lecture notes, ‘New theoretical results in nonequilibrium statistical mechanics‘ [Rue98]; and in the paper of G. Gallavotti, ‘Chaotic dynamics, fluctuations, nonequilibrium ensembles’ [Gal98].

A very beautiful and more advanced discussion of many of the topics covered in the later chapters of this book is provided in the monograph Chaos, Scattering, and Statistical Mechanics [Gas98], by P. Gaspard. Some general reviews of these subjects can also be found in papers by Gaspard; van Beijeren and Dorfman; Cohen; Dellago and Posch; Morriss, Dettmann and Rondoni, in Physica A, 240 Nos. 1–2 (1997), and in the Chaos Focus Issue: Chaos and Irreversibility [TGN98].

Billiard systems

In this book, we have only touched lightly the deep and rich subject of the dynamical theory of hard-sphere systems. This area has been developed by Sinai and co-workers and constitutes one of the most fascinating areas for study – it is a field of beautiful mathematics and of major physical interest.

The definition of a mixing system

While Boltzmann fixed his attention on the motion of the phase point for a single system and was led to the concept of ergodicity, Gibbs took another approach to the same problem. Since one never knows precisely what the initial phase point of a system is, Gibbs decided to consider the average behavior of a set of points on the constant-energy surface with more or less the same macroscopic initial state. Without worrying too much about how such a set might be defined precisely, let's consider an initial set of points A. As the set travels through Γ-space, it changes shape but its measure stays the same, μ{A) = μ(At). The set gets stretched and folded and may eventually appear on a coarse enough scale to fill the energy surface uniformly. However, the set At has the same topological structure as the set A and the initial set is not ‘forgotten’, in the sense that a time-reversal operation on the set At will produce the set A. There is a nice lecture-demonstration apparatus that illustrates this time-reversal operation: A drop of immiscible ink is added to a container of glycerine. If you stir the glycerine slowly, the drop will stretch and form a thin line. Eventually it seems to fill the whole space, but if the stirring is reversed, the initial configuration of the drop of ink surprisingly reappears.

Gibbs thought that the apparently uniform distribution of the set At on the energy surface was the key to understanding how mechanically reversible systems could approach an equilibrium state.