Classical Statistical Mechanics

Section 2.2 of Frenkel & Smit [1] discusses a derivation of the ``quasi-classical'' representation of the canonical partition function, $Q_{\rm classical}$:

$$Q_{\rm classical} = \frac{1}{h^{dN}N!}\int \int d{\bf r}^N d... ...\left[-\beta\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)\right]$$ (42)

$\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)$ is the Hamiltonian function which computes the energy of a point in phase space. The derivation of Eq. 42 is not repeated here. What is important is that the probability of a point in phase space is represented as

$$P\left({\bf r}^N,{\bf p}^N\right) = \left(Q_{\rm classical}\... ...left[-\beta\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)\right].$$ (43)

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:

$$\left<G\right> = \frac{ \mbox{ \begin{minipage}{8cm} \beg... ...,{\bf p}^N\right)\right] \end{displaymath} \end{minipage} }},$$ (44)

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

$$\mathscr{H}\left({\bf r}^N, {\bf p}^N\right) = \mathscr{K}\left({\bf p}^N\right) + \mathscr{U}\left({\bf r}^N\right)$$ (45)

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:

$$Q\left(N,V,T\right) = \frac{1}{h^{3N}N!} \left\{\int d{\bf p}^N \exp\left[-\beta\mathsc... ...int d{\bf r}^N \exp\left[-\beta\mathscr{U}\left({\bf r}^N\right)\right]\right\}$$ (46)

$$= \left\{\frac{V^N}{h^{3N}N!} \int d{\bf p}^N \exp\left[-\beta\math... ...int d{\bf r}^N \exp\left[-\beta\mathscr{U}\left({\bf r}^N\right)\right]\right\}$$ (47)

$$= Q_{ideal} Z$$ (48)

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,

$$\mathscr{K}\left({\bf p}^N\right) = \sum_i \frac{{\bf p}_i^2}{2m_i},$$ (49)

where $m_i$ is the mass of particle $i$, the integral over particle momenta can be evaluated analytically:

$$\int d{\bf p}^N \exp\left[-\beta\mathscr{K}\left({\bf p}^N\right)\right] = \prod_{i=1}{3N}\int dp_i \exp\left(-\frac{p_i^2}{2mk_BT}\right)$$ (50)

$$= \left(2\pi m k_B T\right)^{3N/2}.$$ (51)

(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

$$\frac{V^N}{N!h^{3N}} \int d{\bf p}^N \exp\left[-\beta\mathscr{K}\left({\bf p}^N\right)\right] = \frac{V^N}{N!}\left(\sqrt{\frac{2\pi m k_B T}{h^2}}\right)^{3N}$$ (52)

$$= Q_{ideal}\left(N,V,T\right) = \frac{V^N}{N!\Lambda^{3N}}$$ (53)

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:

$$\left<G\right> = Z^{-1}\int d{\bf r}^N \exp\left[-\beta\mathscr{U}\left({\bf r}^N\right)\right] G\left({\bf r}^N\right).$$ (54)

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

$$\left<G\right> \int d{\bf r}^N G\left({\bf r}^N\right)\rho_{NVT}\left({\bf r}^N\right)$$ (55)

where

$$\rho_{NVT}\left({\bf r}^N\right) \equiv Z^{-1}e^{-\beta U\left({\bf r}^N\right)}$$ (56)

is called the ``canonical probability distribution.'' As pointed out on p. 15 of Frenkel & Smit [1], Eq. 54 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.