Section 2.2 of Frenkel & Smit [1] discusses a derivation of the “quasi-classical” representation of the canonical partition function, \(Q_{\rm classical}\): \begin{equation} \label {eq:q_classical} Q_{\rm classical} = \frac {1}{h^{dN}N!}\int \int d{\bf r}^N d{\bf p}^N \exp \left [-\beta \mathscr {H}\left ({\bf r}^N,{\bf p}^N\right )\right ] \end{equation} \(\mathscr {H}\left ({\bf r}^N,{\bf p}^N\right )\) is the Hamiltonian function which computes the energy of a point in phase space. The derivation of Eq. 35 is not repeated here. What is important is that the probability of a point in phase space is represented as \begin{equation} P\left ({\bf r}^N,{\bf p}^N\right ) = \left (Q_{\rm classical}\right )^{-1} \exp \left [-\beta \mathscr {H}\left ({\bf r}^N,{\bf p}^N\right )\right ]. \end{equation} So, the general “sum-over-states’ ensemble average of quantum statistical mechanics, first presented in Eq. 1, becomes an integral over phase space in classical statistical mechanics: \begin{equation} \label {eq:classical_ensemble_average} \left \langle G\right \rangle = \frac {\int \int d{\bf r}^N d{\bf p}^N \exp \left [-\beta \mathscr {H}\left ({\bf r}^N,{\bf p}^N\right )\right ] G\left ({\bf r}^N,{\bf p}^N\right )}{\int \int d{\bf r}^N d{\bf p}^N \exp \left [-\beta \mathscr {H}\left ({\bf r}^N,{\bf p}^N\right )\right ]}, \end{equation} where \(G\left ({\bf r}^N,{\bf p}^N\right )\) is the value of the observable \(G\) at phase space point \(\left ({\bf r}^N,{\bf p}^N\right )\). Before moving on, it is useful to recognize that we normall simplify this ensemble average by noting that, for a system of classical particles, the usual choice for the Hamiltonian has the form \begin{equation} \label {eq:classical_hamiltonian} \mathscr {H}\left ({\bf r}^N, {\bf p}^N\right ) = \mathscr {K}\left ({\bf p}^N\right ) + \mathscr {U}\left ({\bf r}^N\right ) \end{equation} where \(\mathscr {K}\) is the kinetic energy, which is only a function of momenta, and \(\mathscr {U}\) is the potential energy, which is only a function of position. The canonical partition function, \(Q\), can in this case be factorized: \begin{equation}\label {eq:q_fact} \begin {aligned} Q\left (N,V,T\right ) & = \frac {1}{h^{3N}N!} \left \{\int d{\bf p}^N \exp \left [-\beta \mathscr {K}\left ({\bf p}^N\right )\right ]\right \} \left \{\int d{\bf r}^N \exp \left [-\beta \mathscr {U}\left ({\bf r}^N\right )\right ]\right \} \\ & = \left \{\frac {V^N}{h^{3N}N!} \int d{\bf p}^N \exp \left [-\beta \mathscr {K}\left ({\bf p}^N\right )\right ]\right \} \left \{V^{-N}\int d{\bf r}^N \exp \left [-\beta \mathscr {U}\left ({\bf r}^N\right )\right ]\right \} \\ & = Q_{ideal} Z \end {aligned} \end{equation} The quantity in the left-hand braces is the ideal gas partition function, because it corresponds to the case when the potential \(\mathscr {U}\) is 0. (Note that we have multiplied and divided by \(V^N\); this is the equivalent of scaling the positions in the integration over positions.) The quantity in the right-hand braces is called the configurational partition function, \(Z\).
Because the kinetic energy \(\mathscr {K}\) has the simple form, \begin{equation} \mathscr {K}\left ({\bf p}^N\right ) = \sum _i \frac {{\bf p}_i^2}{2m_i}, \end{equation} where \(m_i\) is the mass of particle \(i\), the integral over particle momenta can be evaluated analytically: \begin{equation} \begin {aligned} \int d{\bf p}^N \exp \left [-\beta \mathscr {K}\left ({\bf p}^N\right )\right ] & = \prod _{i=1}{3N}\int dp_i \exp \left (-\frac {p_i^2}{2mk_BT}\right ) \\ & = \left (2\pi m k_B T\right )^{3N/2}. \end {aligned} \end{equation} (We have assumed all particles have the same mass, \(m\); in the case of distinct masses, this is just a product of similar factors.)
\(Q_{ideal}\) becomes \begin{equation} \begin {aligned} \frac {V^N}{N!h^{3N}} \int d{\bf p}^N \exp \left [-\beta \mathscr {K}\left ({\bf p}^N\right )\right ] & = \frac {V^N}{N!}\left (\sqrt {\frac {2\pi m k_B T}{h^2}}\right )^{3N} \\ = Q_{ideal}\left (N,V,T\right ) & = \frac {V^N}{N!\Lambda ^{3N}} \end {aligned} \end{equation} where \(\Lambda \) is the de Broglie wavelength.
So, when the observable \(G\) is a function of positions only, the ensemble average becomes a configurational average: \begin{equation} \label {eq:config_avg} \left \langle G\right \rangle = Z^{-1}\int d{\bf r}^N \exp \left [-\beta \mathscr {U}\left ({\bf r}^N\right )\right ] G\left ({\bf r}^N\right ). \end{equation} Note that the integation over momentum yields a factor \(Q_{ideal}\) in both the numerator and denominator, and thus divides out. We can write this configurational average using a probability distribution, \(\rho _{NVT}\), as \begin{equation} \left \langle G\right \rangle \int d{\bf r}^N G\left ({\bf r}^N\right )\rho _{NVT}\left ({\bf r}^N\right ) \end{equation} where \begin{equation} \rho _{NVT}\left ({\bf r}^N\right ) \equiv Z^{-1}e^{-\beta U\left ({\bf r}^N\right )} \end{equation} is called the “canonical probability distribution.” As pointed out on p. 15 of Frenkel & Smit [1], Eq. 43 is “the starting point for virtually all classical simulations of many-body systems”; that is, it is the starting point for all simulations discussed in this course.