# Introduction

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:

${\displaystyle {\mathcal {H}}(\mathbb {Q} ,\mathbb {P} )=\sum _{i}{\frac {{\vec {p}}_{i}{}^{2}}{2m_{i}}}+V(\mathbb {Q} )}$
We know that from a microscopic point of view the system can be completely described if we solve Hamilton's equations:
${\displaystyle {\dot {\vec {q}}}_{i}={\frac {\partial {\mathcal {H}}}{\partial {\vec {p}}_{i}}}={\frac {{\vec {p}}_{i}}{m_{i}}}\quad \qquad {\dot {\vec {p}}}_{i}=-{\frac {\partial {\mathcal {H}}}{\partial {\vec {q}}_{i}}}=-{\frac {\partial V}{\partial {\vec {q}}_{i}}}}$
with given initial conditions that we call ${\displaystyle (\mathbb {Q} _{0},\mathbb {P} _{0})}$, so that in the end we know ${\displaystyle (\mathbb {Q} (t),\mathbb {P} (t))}$ for any ${\displaystyle t}$. If we now call ${\displaystyle O(\mathbb {Q} ,\mathbb {P} )}$ a generic observable of the system we can define its time average as:
${\displaystyle {\overline {O}}(\mathbb {Q} _{0},\mathbb {P} _{0})=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}O(\mathbb {Q} (t),\mathbb {P} (t))dt}$
We can therefore ask ourselves: is there a relation between the time average ${\displaystyle {\overline {O}}}$ and the ensemble average ${\displaystyle \left\langle O\right\rangle }$ of the observable ${\displaystyle O}$? 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 ${\displaystyle {\overline {O}}}$, as can be seen from its definition, derives solely from the microscopic properties of the system while ${\displaystyle \left\langle O\right\rangle }$ 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 ${\displaystyle {\overline {O}}=\left\langle O\right\rangle }$ but unfortunately to date it is still not known when these hypotheses do apply.