Editing Microcanonical ensemble (section)

== Examples ==
=== 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>

=== Ideal gas in a uniform gravitational field ===
The microcanonical phase volume can also be calculated explicitly for an ideal gas in a uniform [[gravitational field]].<ref name="RomanWhite1995">{{cite journal|last1=Roman|first1=F L|last2=White|first2=J A|last3=Velasco|first3=S|title=Microcanonical single-particle distributions for an ideal gas in a gravitational field|journal=European Journal of Physics|volume=16|issue=2|year=1995|pages=83–90|issn=0143-0807|doi=10.1088/0143-0807/16/2/008|bibcode=1995EJPh...16...83R |s2cid = 250840083 }}</ref>

The results are stated below for a 3-dimensional ideal gas of <math>N</math> particles, each with mass <math>m</math>, confined in a thermally isolated container that is infinitely long in the ''z''-direction and has constant cross-sectional area <math>A</math>. The gravitational field is assumed to act in the minus ''z'' direction with strength <math>g</math>. The phase volume <math>W(E, N)</math> is
<math display="block">
W(E, N) = \frac{(2 \pi)^{3N/2} A^N m^{N/2}}{g^N \, \Gamma{\left(\frac{5N}{2}\right)}} E^{\frac{5N}{2}-1}
</math>
where <math>E</math> is the total energy, kinetic plus gravitational.

The gas density <math>\rho(z)</math> as a function of height <math>z</math> can be obtained by integrating over the phase volume coordinates. The result is:
<math display="block">
\rho(z) = \left(\frac{5N}{2} - 1 \right) \frac{mg}{E}\left(1 - \frac{mgz}{E} \right)^{\frac{5N}{2}-2}
</math>
Similarly, the distribution of the velocity magnitude <math>\left|\mathbf{v}\right|</math> (averaged over all heights) is
<math display="block">
f(|\mathbf{v}|)
= \frac{\Gamma{\left(\frac{5N}{2}\right)}}{\Gamma{\left(\frac{3}{2}\right)} \, \Gamma{\left(\frac{5N}{2} - \frac{3}{2}\right)}}
\cdot \frac{m^{3/2} {\left|\mathbf{v}\right|}^2}{2^{1/2} E^{3/2}}
\cdot \left(1 - \frac{m {\left|\mathbf{v}\right|}^2}{2 E} \right)^{{5(N-1)}/{2}}
</math>
The analogues of these equations in the canonical ensemble are the [[barometric formula]] and the [[Maxwell–Boltzmann distribution]], respectively. In the limit <math>N \to \infty</math>, the microcanonical and canonical expressions coincide; however, they differ for finite <math>N</math>. In particular, in the microcanonical ensemble, the positions and velocities are not statistically independent. As a result, the kinetic temperature, defined as the average kinetic energy in a given volume <math>A \, dz</math>, is nonuniform throughout the container:
<math display="block">
T_{\text{kinetic}} = \frac{3 E}{5 N - 2} \left(1 - \frac{mgz}{E} \right)
</math>
By contrast, the temperature is uniform in the canonical ensemble, for any <math>N</math>.<ref name="VelascoRomán1996">{{cite journal|last1=Velasco|first1=S|last2=Román|first2=F L|last3=White|first3=J A|title=On a paradox concerning the temperature distribution of an ideal gas in a gravitational field|journal=European Journal of Physics|volume=17|issue=1|year=1996|pages=43–44|issn=0143-0807|doi=10.1088/0143-0807/17/1/008|s2cid=250885860}}</ref>