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Two of the most common approaches to statistical mechanics start with one of the following principles.
Boltzmann equation
The Boltzmann equation for entropy \(S\),
\begin{equation}
S = k\ln \Omega
\end{equation}
in terms of number of macro states, $\Omega$, of an isolated system, is the most important equation of statistical mechanics. This along with principle of maximum entropy forms basis of all equilibrium statistical mechanics.
Postulate of equal a priori probabilities
Total energy is constant in time. There are several micro states with a given energy and micro state is changing with time. Suppose we know energy and volume of the system, most reasonable assumption to make is
All micro states of a system that have the same energy are assumed to be equally probable.
This is the basic assumption of statistical mechanics and is known as the axiom of {\tt equal a priori probabilities}. Correctness, or otherwise, of this assumption is to be checked against experiments. There is no evidence pointing that this assumption is not correct. The assumption of equal a priori probabilities of micro states along with the requirement that for system in equilibrium the probability must be a maximum give another starting point of statistical mechanics.
One of the two above statements can be taken as the fundamental principle of statistical mechanic.
