4 - Foundation of statistical mechanics and dynamical systems

Summary

To know that you know when you do know, and know that you do not know when you do not know: that is knowledge.

Confucius

Statistical mechanics was founded by Maxwell, Boltzmann and Gibbs to account for the properties of macroscopic bodies, systems with a very large number of particles, without very precise requirements on the dynamics (except for the assumption of ergodicity).

Since the discovery of deterministic chaos it is now well established that statistical approaches may also be unavoidable and useful, as discussed in Chapter 1, in systems with few degrees of freedom. However, even after many years there is no general agreement among the experts about the fundamental ingredients for the validity of statistical mechanics.

It is quite impossible in a few pages to describe the wide spectrum of positions ranging from the belief of Landau and Khinchin in the main role of the many degrees of freedom and the (almost) complete irrelevance of dynamical properties, in particular ergodicity, to the opinion of those, for example Prigogine and his school, who consider chaos as the basic ingredient.

For almost all practical purposes one can say that the whole subject of statistical mechanics consists in the evaluation of a few suitable quantities (for example, the partition function, free energy, correlation functions). The ergodic problem is often forgotten and the (so-called) Gibbs approach is accepted because “it works.” Such a point of view cannot be satisfactory, at least if one believes that it is not less important to understand the foundation of such a complex issue than to calculate useful quantities.