The foundations of statistical mechanics
Before proceeding to the study of the canonical and grand canonical ensembles, let us stop for a moment to reflect on the deep reasons that led us to the formulation of the microcanonical ensemble. This is extremely important since as we shall later see both the canonical and the grand canonical ensembles are based upon the microcanonical one, so understanding why the microcanonical ensemble is the way it is can help us determining the foundations of statistical mechanics.
In particular one could note that the way we have introduced the microcanonical ensemble is not really "philosophically" satisfying, particularly because of the a priori equal probability postulate. We could in fact ask ourselves: is this postulate realistic? In other words: are all the microstates of any given system really equally probable, or can we find cases where this is not true? And if they are really equally probable, why is it so?
The topic is very wide and complicated (in its deepest nature it is still an open problem), and since it is not essential in order to understand the rest of this chapter we postpone its study in the appendix A more convincing foundation of statistical mechanics .