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).