[NOTES/SM-03004] Application of Micro Canonical Ensemble to Ideal Gas

Consider a system of ideal gas of \(N\) molecules in a volume \(V\)and having energy
\begin{equation}
E =\sum_{k=1}^{3N} \frac{p_k^2}{2m}.
\end{equation}
The number of microstates of the ideal gas are given by
\begin{equation}
\Omega = \frac{1}{N!}\frac{V^N}{h^{3N}}
\frac{(2\pi mE)^{3N/2}}{\Gamma\big(\frac{3N}{2}+1)}.
\end{equation}
Therefore
\begin{equation}
\log \Omega = N \log V + \frac{3N}{2}V -\log N! -\log 3N \log h -\log \Gamma\big(\frac{3N}{2}+1).
\end{equation}
The micro canonical entropy is given by
\begin{equation}
S(E,V,N)= k_B \log \Omega(E,V,N)
\end{equation}
Compute pressure and temperature

\begin{eqnarray}\frac{1}{T} &=& \Big(\pp[S(E,V,N]{E} \Big)_{V,N} = \frac{3}{2} \frac{Nk_B}{E}\\
\frac{P}{T} &=& \Big(\pp[S(E,V,N]{V} \Big)_{E,N} =  \frac{N k_B}{V}.
\end{eqnarray}
Thus we get the ideal gas law
\begin{equation}
PV=Nk_BT
\end{equation}
and the law of equipartition of energy
\begin{equation}
E = 3N(k_BT/2)
\end{equation}