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.

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.