Editing Maxwell–Boltzmann statistics (section)

==Derivation from canonical ensemble==

The Maxwell-Boltzmann distribution describes the probability of a particle occupying an energy state ''E'' in a classical system. It takes the following form:

:<math>
\begin{align}
f_{\text{MB,high}}(E) &=\exp\left(-\frac{E - E_F}{kT}\right),  &\text{for } E \gg E_F \\ 
f_{\text{MB,low}}(E) &=1-\exp\left(\frac{E - E_F}{kT}\right),  &\text{for } E \ll E_F
\end{align}
</math>

For a system of indistinguishable particles, we start with the canonical ensemble formalism.

In a system with energy levels <math>\{E_i\}</math>, let <math>n_i</math> be the number of particles in state ''i''. The total energy and particle number are:
:<math>
\begin{align}
E_{\text{total}} &= \sum_i n_i E_i \\
N &= \sum_i n_i
\end{align}
</math>

For a specific configuration <math>\{n_i\}</math>, the probability in the canonical ensemble is:
:<math>P(\{n_i\}) = \frac{1}{Z_N} \frac{N!}{\prod_i n_i!} \prod_i (e^{-\beta E_i})^{n_i}</math>

The factor <math>\frac{N!}{\prod_i n_i!}</math> accounts for the number of ways to distribute ''N'' indistinguishable particles among the states.

For Maxwell-Boltzmann statistics, we assume that the average occupation number of any state is much less than 1 (<math>\langle n_i \rangle \ll 1</math>), which leads to:
:<math>\langle n_i \rangle \approx e^{-\beta(E_i-\mu)}</math>

where <math>\mu</math> is the chemical potential determined by <math>\sum_i \langle n_i \rangle = N</math>.

For energy states near the Fermi energy <math>E_F</math>, we can express <math>\mu \approx E_F</math>, giving:
:<math>f_{\text{MB}}(E) = e^{-(E-E_F)/kT}</math>

For high energies (<math>E \gg E_F</math>), this directly gives:
:<math>f_{\text{MB,high}}(E) = e^{-(E-E_F)/kT}</math>

For low energies (<math>E \ll E_F</math>), using the approximation <math>e^{-x} \approx 1-x</math> for small ''x'':
:<math>f_{\text{MB,low}}(E) \approx 1 - e^{(E-E_F)/kT}</math>

This is the derivation of the Maxwell-Boltzmann distribution in both energy regimes.