Equation of state of a classical electron layer

The equation of state of a dilute two-dimensional classical electron gas is calculated by the virial expansion in the ring approximation. Four corrections to ideal-gas behavior are obtained exactly; the central result, in terms of the small parameter ${\epsilon }^2=\frac{4\pi \rho e^4}{{\left(k_{B}T\right)}^2}$, is $\frac{P}{\rho k_{B}T}-1=\frac{{\epsilon }^2\ln \left(2\mathrm{}\epsilon \right)}{4}-\frac{(1-\frac{\gamma }{2}){\epsilon }^2}{4}-\frac{{\epsilon }^4{\ln }^2\left(\frac{\epsilon }{2}\right)}{4}+\frac{(\frac{1}{2}-\gamma ){\epsilon }^4\ln \epsilon }{2}+O\left({\epsilon }^4\right)$, where $\gamma$ is Euler's constant. The lowest-order correction agrees with a Debye-Hückel calculation, which is also presented.