Classical Statistical Mechanics

Analogous to the quasi-classical microcanonical paritition function of Eq. 6, here is the quasi-classical representation of the canonical partition function:

$$Q_{\rm classical} = \frac{1}{h^{dN}N!}\int \int d{\bf r}^N d{\bf p}^N \exp\left[-\beta\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)\right]$$ (46)

$\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)$ is the Hamiltonian function which computes the energy of a point in phase space. The probability of a point in phase space is represented as

$$P\left({\bf r}^N,{\bf p}^N\right) = \left(Q_{\rm classical}\right)^{-1} \exp\left[-\beta\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)\right].$$ (47)

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:

$$\left<G\right> = \frac{\displaystyle \int \int d{\bf r}^N d{\bf p... ...{\bf p}^N \exp\left[-\beta\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)\right] },$$ (48)

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 normally simplify this ensemble average by noting that, for a system of classical particles, the usual choice for the Hamiltonian has the form

$$\mathscr{H}\left({\bf r}^N, {\bf p}^N\right) = \mathscr{K}\left({\bf p}^N\right) + U\left({\bf r}^N\right)$$ (49)

where $\mathscr{K}$ is the kinetic energy, which is only a function of momenta, and $U$ is the potential energy, which is only a function of position. The canonical partition function, $Q$, can in this case be factorized:

$$Q\left(N,V,T\right) = \frac{1}{h^{3N}N!} \left\{\int d{\bf p}^N \exp\left[-\beta\mathsc... ... \left\{\int d{\bf r}^N \exp\left[-\beta U\left({\bf r}^N\right)\right]\right\}$$ (50)

$$= \left\{\frac{V^N}{h^{3N}N!} \int d{\bf p}^N \exp\left[-\beta\math... ...\{V^{-N}\int d{\bf r}^N \exp\left[-\beta U\left({\bf r}^N\right)\right]\right\}$$ (51)

$$= Q_{\rm ideal} Z$$ (52)

The quantity in the left-hand braces is the ideal gas partition function, because it corresponds to the case when the potential $U$ 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,

$$\mathscr{K}\left({\bf p}^N\right) = \sum_i \frac{{\bf p}_i^2}{2m_i},$$ (53)

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

$$\int d{\bf p}^N \exp\left[-\beta\mathscr{K}\left({\bf p}^N\right)\right] = \prod_{i=1}^{3N}\int dp_i \exp\left(-\frac{p_i^2}{2mk_BT}\right)$$ (54)

$$= \left(2\pi m k_B T\right)^{3N/2}.$$ (55)

(We have assumed all particles have the same mass, $m$; in the case of distinct masses, this is just a product of similar factors.)

$Q_{\rm ideal}$ becomes

$$\frac{V^N}{N!h^{3N}} \int d{\bf p}^N \exp\left[-\beta\mathscr{K}\left({\bf p}^N\right)\right] = \frac{V^N}{N!}\left(\sqrt{\frac{2\pi m k_B T}{h^2}}\right)^{3N}$$ (56)

$$= Q_{\rm ideal}\left(N,V,T\right) = \frac{V^N}{N!\Lambda^{3N}}$$ (57)

where $\Lambda$ is the de Broglie wavelength, a quantum-mechanical property of a particle inversely proportional to its momentum (and thus inversely proportional to the square root of temperature):

$$\Lambda = \sqrt{\frac{h^2}{2\pi m k_BT}}$$ (58)

As an example, for a hydrogen atom with mass 1 amu and at room temperature (298 K), $\Lambda\approx$ 1 Å. The de Broglie wavelength limits the precision by which a particle's position can be determined; for H atoms at room temperature, one is not permitted to specify their positions with a precision finer than about 1 ångstrom without violating the Heisenberg uncertainty principle of quantum mechanics. However, as we will see, in classical molecular simulations, we must lift this restriction, while never forgetting that this makes a classical representation of a molecule somewhat less realistic.

With the momentum degrees of freedom handled at finite temperature, when the observable $G$ is a function of positions only, the ensemble average becomes a configurational average:

$$\left<G\right> = Z^{-1}\int d{\bf r}^N \exp\left[-\beta U\left({\bf r}^N\right)\right] G\left({\bf r}^N\right).$$ (59)

Note that the integration over momentum yields a factor $Q_{\rm ideal}$ in both the numerator and denominator, and thus divides out. We can write this configurational average using a probability distribution, $\rho_{NVT}$, as

$$\left<G\right> = \int d{\bf r}^N G\left({\bf r}^N\right)\rho_{NVT}\left({\bf r}^N\right)$$ (60)

where

$$\rho_{NVT}\left({\bf r}^N\right) \equiv Z^{-1}e^{-\beta U\left({\bf r}^N\right)}$$ (61)

is called the “canonical probability distribution.” As pointed out on p. 15 of Frenkel & Smit [1], Eq. 59 is “the starting point for virtually all classical simulations of many-body systems”; that is, it is the starting point for almost all simulations discussed in this course.