3.Formal derivation of the virial expansion

The expression for the grand canonical partition function

$$Y (T,V,\mu) = 1+\sum_{N=1}^\infty e^{\beta \mu N} Z(T,V,N)$$ (56)

$$= 1 + \sum_{N=1}^\infty \frac{e^{\beta \mu N}}{N!} \frac{1}{\lambda_T^{3 N}} Q_N (T,V) ,$$

where

$$Q_N(T,V) = \int d^{3N} x \exp (-\beta {\cal{H}}_{\rm {pot}})$$

is the configuration integral for the $N$-particle system, can be considered as a TAYLOR expansion with respect to $\exp (\beta \mu)$, which is called the fugacity. We know that for a gas of low density $\exp (\beta \mu)$ is very small. Since the whole sum must converge in any case, this TAYLOR series must converge, too. So far no problem.

For $N=1$ and 2 the configuration integrals are given by

$$Q_1 = V$$ (57)

$$Q_2 = \int \!\!\!\int\limits_{(V)} d^3 r_1 d^3 r_2 \left( \underbrace{e^{-\beta V (r_{12})}-1}_{=f(r_{12})} + 1 \right)$$

$$= V \int d^3 r f(r) + V^2 .$$ (58)

In the derivation of expression (58) we took into account that the Mayer $f$-function is appreciably different from zero only within the range of the interatomic potential $V(r)$, which is negligibly small compared with the dimension $V^{1/3}$ of the system.

The quantity with a physical meaning is the grand canonical potential

$$J(T,V, \mu) = -k_B T \ln Y (T,V, \mu) ,$$ (59)

not the partition function. Inserting (56) into (59) and rearranging, we obtain $J$ as a TAYLOR series in $\exp (\beta \mu)$. To see what happens, we calculate just the first two terms of this series. According to

$$\ln (1+x) = x - \frac{x^2}{2} + 0(x^3)$$ (60)

we find

$$\ln Y = e^{\beta \mu} V/\lambda_T^3 + \frac{1}{2} e^{2 \beta \mu} Q_2/\lambda_T^6 + \cdots$$

$$- \frac{1}{2} \left( e^{\beta \mu} V/\lambda_T^3 + \cdots \right)^2$$

$$+ \cdots$$

$$= e^{\beta \mu} V/\lambda_T^3 + \frac{1}{2} e^{2 \beta \mu} (Q_2 - V^2) / \lambda_T^6 + \cdots$$

$$= V\left\{ e^{\beta \mu}/\lambda_T^3 + \frac{1}{2} \left( e^{\beta \mu} / \lambda_T^3\right)^2 \left( \frac{Q_2}{V} - V\right) + \cdots \right\}$$ (61)

From $J$ we obtain pressure $P$ and mean particle number $\overline N$ as

$$P = - \frac{J}{V} = k_B T \left\{ e^{\beta \mu}/\lambda_T^3 + \frac{1... ... \mu} / \lambda_T^3 \right)^2 \left( \frac{Q_2}{V} - V\right) + \cdots \right\}$$ (62)

$$\overline N = - \left( \frac{\partial J}{\partial \mu}\right)_{T,V} = V \left\{... ... \mu}/ \lambda_T^3 \right)^2 \left( \frac{Q_2}{V} - V \right) + \cdots \right\}$$ (63)

Inverting the expansion (63) of $n=\overline N/V$ with respect to $\exp (\beta \mu)$, we obtain the fugacity $\exp (\beta \mu)$ as a power series in $n$:

$$e^{\beta \mu} / \lambda_T^3 = n- \left( \frac{Q_2}{V} - V\right) n^2 + \cdots .$$ (64)

Inserting this power series (64) into (62) and again rearranging, we obtain the pressure $P$ as a power series in the particle density $n$, i.e. the virial expansion. Up to second order we find from (64) and (62)

$$P = k_B T \left\{ n- \frac{1}{2} n^2 \left( \frac{Q_2}{V} - V \right) + \cdots \right\} ,$$ (65)

from which we identify, using (58), the second virial coefficient as

$$B(T) = - \frac{1}{2} \int d^3 r f (r) .$$ (66)

There is a startling feature in this seemingly elegant derivation of the virial expansion. It concerns the convergence of the fugacity expansion of the grand canonical partition function (56), our starting point. For dilute gases, for which according to (64)

$$e^{\beta \mu} / \lambda_T^3 \approx n$$ (67)

holds, we obtain the first three terms of this expansion as

$$Y \approx 1+N+N^2/2 + \cdots$$ (68)

These are the first terms of the TAYLOR expansion of $\exp N$, which for our large values of $N$ converges extremely slowly. It would be ridiculous to consider the first three terms (68) as an approximation for the value of $Y$. However, expanding the logarithm of $Y$ and rearranging miraculously produces a rapidly convergent series for $J$ (or $P$). The explanation of this miracle is that $Y$ is not just any function, but is given by

$$Y=e^{\beta P V} ,$$ (69)

where the pressure $P$ is an intensive quantity (i.e. independent of $N$ for fixed density $n$). By the rearrangement which leads to expression (61) for $\ln Y$, we effectively pick all the extensive terms (i.e. proportional to $N$ at fixed $n$) in the expansion (56) for $Y$. In the following note it is shown that $\ln Y$ is indeed extensive and has a fugacity expansion with coefficients given by integrals over clusters of interacting particles.