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.