[[Statistical thermodynamics MOC]]
# Modified Boltzmann approximation

**Modified Boltzmann statistics** is an approximation applicable to systems of $N$ identical and indistinguishable particles at a high temperature with many accessible microstates so that it is unlikely for any two particles to be in the same quantum state.
The [[Canonical ensemble|canonical partition function]] is given by
$$
\begin{align*}
Z \approx \frac{\tilde{Z}}{N!} = \frac{(Z_{1})^N}{N!}
\end{align*}
$$
where $\tilde{Z}$ is the canonical partition function given by [[Maxwell-Boltzmann statistics]], and
$$
\begin{align*}
Z_{1} = \sum_{i} \exp\left( -\frac{\varepsilon_{i}}{k_{B}T} \right)
\end{align*}
$$
is the canonical partition function for a single particle.
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