17.3: The Average Ensemble Energy is Equal to the Observed Energy of a System
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 \( \langle E \rangle \).