Introduction to Equilibrium Statistical Mechanics

Abstract

The only microscopic description of thermodynamic systems that we have seen so far is the one provided by kinetic theory. That of kinetic theory, is an approach applicable in principle to a great variety of systems, both in equilibrium and in non-equilibrium conditions, but only at the price of closure approximations on the many-particle statistics, that are often difficult to control.

Appendices

Appendix

A.1 Perturbation Methods

The calculation of the partition function for an ideal gas, that we carried out in Sect. 5.4.1, ended up consisting of an evaluation of Gaussian integrals. In the presence of interactions, however, the Hamiltonian $\fancyscript{H}(\Gamma )$ will contain non-quadratic terms that make the evaluation of the integrals in Eq. (5.29) not possible, in general, in closed form. Approximation methods become then necessary.

Two possibilities that become immediately apparent, from inspection of Eq. (5.29), are high temperature and low temperature expension of the integrand. The first case corresponds to a regular expansion in the interaction term of the Hamiltonian, using the factor $1/T$ as expansion parameter. The second case corresponds to a saddle point (Laplace) kind of expansion of the integral. We illustrate the two procedures on two specific examples.

5.1.1 A.1.1 High Temperature Expansion

We consider the case of real gas, in which molecules interact with a two-body potential:

$$\begin{aligned} \fancyscript{H}=\frac{1}{2m}\sum _ip_i^2+\sum _{i\ne j}U(\mathbf{q}_i-\mathbf{q}_j). \end{aligned}$$

(5.60)

The expansion of Eq. (5.29) is carried out with respect to the interaction term $U$:

$$\begin{aligned} Z&=\frac{1}{N!\delta \Gamma } \int \text {d}^{3N}p\text {d}^{3N}q\ \exp \Big (-\frac{1}{2mT}\sum _ip_i^2\Big )\Big [1-\frac{1}{T} \sum _{i\ne j}U(\mathbf{q}_i-\mathbf{q}_j)\nonumber \\&\quad +\frac{1}{2T^2}\sum _{i\ne j}\sum _{k\ne l} U(\mathbf{q}_i-\mathbf{q}_j)U(\mathbf{q}_k-\mathbf{q}_l)+\ldots \Big ]. \end{aligned}$$

(5.61)

Noticing that the kinetic part decouples from the interaction part in the integral, Eq. (5.61), can be rewritten in the form

$$\begin{aligned} Z&=Z_0\Big [1-\frac{N(N-1)}{TV^2}\int \text {d}^3q_1\text {d}^3q_2\ U(\mathbf{q}_1-\mathbf{q}_2)+\ldots \Big ],\nonumber \\&=Z_0\Big [1-\frac{N}{Tv}\int \text {d}^3q\ U(\mathbf{q})+\ldots \Big ], \end{aligned}$$

(5.62)

where $v=V/N$ is the specific volume of one molecule and

$$\begin{aligned} Z_0=\frac{V^N}{N!\delta \Gamma }\int \text {d}^{3N}p\ \exp \Big (-\frac{1}{2mT}\sum _ip_i^2\Big ) \end{aligned}$$

(5.63)

is the expression of the partition function in the ideal case, already provided in Eq. (5.31).

We call attention to the structure of the expansion in Eq. (5.61), that appears to be, essentially, an expansion in the interaction length of the potential. To lowest order, $Z=Z_0$, the particles become totally independent, as the effect of interaction is killed by thermal motion. To next order, we have a linear correction, that is the contribution from the average potential energy of the molecules, in the field of the other molecules in the gas. No molecule correlations are taken into account at this level. Proceeding in the perturbation expansion, clusters involving larger number of particles must be taken into account, hence the name for this perturbative approach, of cluster expansion . These increasingly more complex clusters account for the way in which higher order correlations contribute to the dynamics, as the interaction becomes stronger.

5.1.2 A.1.2 Low Temperature Expansion

For $T\rightarrow 0$, the system will tend to lie at the lowest possible energy levels available to the dynamics. As temperature decreases, also the fluctuations around the zero energy will become smaller. This again simplifies life in the case the Hamiltonian can be expressed as a quadratic free component, plus a higher order interaction term. Let us consider for simplicity the case of a one dimensional enharmonic oscillator:

$$\begin{aligned} \fancyscript{H}=\frac{p^2}{2m}+\frac{\alpha q^2}{2}+\lambda q^4, \end{aligned}$$

(5.64)

with $\alpha$ and $\lambda$ positive to make the dynamics finite, and have a single potential minimum at $q=0$. The Hamiltonian in Eq. (5.64) could model e.g. a mesoscopic system in contact with a thermal bath, for which fluctuations are small but not negligible. The thermodynamics of the system will be described again by the partition function

$$\begin{aligned} Z=(\delta \Gamma )^{-1} \int \text {d}p\text {d}q\ \exp \Big (-\frac{1}{T}\Big (\frac{p^2}{2m}+\frac{\alpha q^2}{2}+\lambda q^4\Big )\Big ) \end{aligned}$$

(5.65)

The difficult part of the evaluation of $Z$ lies in the quartic term $\lambda q^4$, that makes the integral non-Gaussian. Nevertheless, for $T\rightarrow 0$, we expect fluctuations to be smaller, and the quartic term to become a correction. This condition can be made explicit, rescaling the coordinate part of the partition function

$$ q\rightarrow \hat{q}=(\alpha /T)^{1/2}q. $$

Substituting into Eq. (5.65), after carrying out the integral over the momentum:

$$\begin{aligned} Z=\frac{(\pi m)^{1/2}T}{\alpha ^{1/2}\delta \Gamma } \int \text {d}\hat{q}\exp \Big (-\frac{\hat{q}^2}{2}+\hat{\lambda }\hat{q}^4\Big ), \qquad \hat{\lambda }=\frac{T^2\lambda }{\alpha ^2}. \end{aligned}$$

(5.66)

We thus see that the low temperature expansion converts into an expansion in the (rescaled) coupling constant $\hat{\lambda }$:

$$\begin{aligned} Z=\frac{(\pi m)^{1/2}T}{\alpha ^{1/2}\delta \Gamma } \int \text {d}\hat{q}\exp \Big (-\frac{\hat{q}^2}{2}\Big )\Big [1+\hat{\lambda }\hat{q}^4+\frac{1}{2} \hat{\lambda }^2\hat{q}^8+\cdots \Big ]. \end{aligned}$$

(5.67)