Editing Microcanonical ensemble (section)

=== Ideal gas ===
The fundamental quantity in the microcanonical ensemble is <math>W(E, V, N)</math>, which is equal to the phase space volume compatible with given <math>(E, V, N)</math>. From <math>W</math>, all thermodynamic quantities can be calculated. For an [[ideal gas]], the energy is independent of the particle positions, which therefore contribute a factor of <math>V^N</math> to <math>W</math>. The momenta, by contrast, are constrained to a <math>3N</math>-dimensional [[n-sphere|(hyper-)spherical]] shell of radius <math>\sqrt{2mE}</math>; their contribution is equal to the surface volume of this shell. The resulting expression for <math>W</math> is:<ref name="Kardar2007">{{cite book|title=Statistical Physics of Particles|first=Mehran|last=Kardar|author-link=Mehran Kardar|publisher=Cambridge University Press|year=2007|isbn=978-0-521-87342-0|pages=105–109}}</ref>
<math display="block"> 
W = \frac{V^N}{N!} \frac{2\pi^{3N/2}}{\Gamma(3N/2)}\left(2mE\right)^{(3N-1)/2}
</math>
where <math> \Gamma(\cdot) </math> is the [[gamma function]], and the factor <math>N!</math> has been included to account for the [[Identical particles|indistinguishability of particles]] (see [[Gibbs paradox]]). In the large <math>N</math> limit, the Boltzmann entropy <math>S = k_{\mathrm{B}} \log W</math> is
<math display="block">
S = k_\text{B} N \log \left[ \frac VN \left(\frac{4\pi m}{3}\frac EN\right)^{3/2}\right] + \frac{5}{2} k_\text{B} N + O\left( \log N \right)
</math>
This is also known as the [[Sackur–Tetrode equation]].

The temperature is given by
<math display="block">
\frac{1}{T} \equiv \frac{\partial S}{\partial E} = \frac{3}{2} \frac{N k_\text{B}}{E}
</math>
which agrees with analogous result from the [[kinetic theory of gases]]. Calculating the pressure gives the [[ideal gas law]]:
<math display="block">
\frac{p}{T} \equiv \frac{\partial S}{\partial V} = \frac{N k_\text{B}}{V} \quad \rightarrow \quad pV = N k_\text{B} T
</math>
Finally, the [[chemical potential]] <math>\mu</math> is
<math display="block">
\mu \equiv -T \frac{\partial S}{\partial N} = -k_\text{B} T \log \left[\frac{V}{N} \, \left(\frac{4 \pi m E}{3N} \right)^{3/2} \right]
</math>