In Chapter 4 we discussed the connection between chaos and equilibrium statistical mechanics, in particular with respect to the ergodic hypothesis. We saw that in systems with many degrees of freedom, chaos (in the sense of at least one positive Lyapunov exponent) is not strictly necessary (nor sufficient) to obtain good agreement between the time average and phase average. This is due, as Boltzmann himself thought and Khinchin proved for an ideal gas, to the fact that in systems with many components, for a large class of observables, the validity of the ergodic hypothesis is basically a consequence of the law of large numbers, and it has a rather weak connection with the underlying dynamics.

From a conceptual point of view the ergodic approach (possibly in a “weak” variant, only pertaining to some interesting macroscopic variables) can be seen as a natural way to introduce probabilistic concepts in a deterministic context. In addition, since one deals with a unique system (although with many degrees of freedom) the ergodicity is a possible (unique?) way to found the equilibrium statistical mechanics on a physical ground, i.e. by exploiting the frequentistic interpretation of probability to extract a statistical description from the analysis of a single experimental trajectory. Finally, on the basis of the results in Chapter 4, and not forgetting that thermodynamics, as a physical theory, was developed to describe the properties of single systems made of many microscopic interacting parts, it seems to us that it is quite fair to conclude that the ensemble viewpoint is just a useful mathematical tool.