As we have stated in The foundations of statistical mechanics, we would like to understand if a more rigorous foundation of statistical mechanics can be established. In particular we would like to see if the equal a priori probability in the phase space of a system is a general property of any physical system, and if so why does it occur.

To make things more tangible let us consider a system with Hamiltonian:

We know that from a microscopic point of view the system can be completely described if we solve Hamilton's equations:
with given initial conditions that we call , so that in the end we know for any . If we now call a generic observable of the system we can define its time average as:
We can therefore ask ourselves: is there a relation between the time average and the ensemble average of the observable ? And in particular are there any hypotheses under which these two averages are equal? It is very important to find an answer to this question because , as can be seen from its definition, derives solely from the microscopic properties of the system while comes from its statistical description given within the microcanonical ensemble. Therefore if we can find a relation between these two quantities we can establish a microscopic foundation of the microcanonical ensemble and thus of the whole statistical mechanics.

As we will see, we will be able to determine under which hypotheses but unfortunately to date it is still not known when these hypotheses do apply.