We will be restricting ourselves to the canonical ensemble (constant temperature and constant pressure). Consider a collection of \(N\) molecules. The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). That is, in a collection of \(N\) molecules, the probability of the molecules having energy \(E_i\): \[P_i = \dfrac{n_i}{N} \nonumber \] This is the directly obtained from the Boltzmann distribution, where the fraction of molecules \(n_i /N\) having energy \(E_i\) is: \[P_i = \dfrac{n_i}{N} = \dfrac{e^{-E_i/kT}}{Q} \label{BD1} \] The average energy is obtained by multiplying \(E_i\) with its probability and summing over all \(i\): \[ \langle E \rangle = \sum_i E_i P_i \label{Mean1} \] Equation \(\ref{Mean1}\) is the standard average over a distribution commonly found in quantum mechanics as expectation values . The quantum mechanical version of this Equation is \[ \langle \psi | \hat{H} | \psi \rangle \nonumber \] where \(\Psi^2\) is the distribution function that the Hamiltonian operator (e.g., energy) is averaged over; this equation is also the starting point in the Variational method approximation. Equation \(\ref{Mean1}\) can be solved by plugging in the Boltzmann distribution (Equation \(\ref{BD1}\)): \[ \langle E \rangle = \sum_i{ \dfrac{E_ie^{-E_i/ kT}}{Q}} \label{Eq1} \] Where \(Q\) is the partition function: \[ Q = \sum_i{e^{-\dfrac{E_i}{kT}}} \nonumber \] We can take the derivative of \(\ln{Q}\) with respect to temperature, \(T\): \[ \left(\dfrac{\partial \ln{Q}}{\partial T}\right) = \dfrac{1}{kT^2}\sum_i{\dfrac{E_i e^{-E_i/kT}}{Q}} \label{Eq2} \] Comparing Equation \(\ref{Eq1}\) with \(\ref{Eq2}\), we obtain: \[ \langle E \rangle = kT^2 \left(\dfrac{\partial \ln{Q}}{\partial T}\right) \nonumber \] It is common to write these equations in terms of \(\beta\), where: \[ \beta = \dfrac{1}{kT} \nonumber \] The partition function becomes: \[ Q = \sum_i{e^{-\beta E_i}} \nonumber \] We can take the derivative of \(\ln{Q}\) with respect to \(\beta\): \[ \left(\dfrac{\partial \ln{Q}}{\partial\beta}\right) = -\sum_i{\dfrac{E_i e^{-\beta E_i}}{Q}} \nonumber \] And obtain: \[ \langle E \rangle = -\left(\dfrac{\partial \ln{Q}}{\partial\beta}\right) \nonumber \] Replacing \(1/kT\) with \(\beta\) often simplifies the math and is easier to use. It is not uncommon to find the notation changes: \(Z\) instead of \(Q\) and \(\bar{E}\) instead of \( \langle E \rangle \).