Equilibrium properties of a two-dimensional Coulomb gas

The canonical partition function of the two-dimensional Coulomb gas interacting through the Coulomb potential ---${q}_{i}{q}_{j}\mathrm{In}{r}_{\mathrm{ij}}$ is considered in detail. The equation of state $\frac{\mathrm{PV}}{2N}={k}_{B}T\ensuremath{-}\frac{{q}^{2}}{4}$ is shown to be meaningful above the critical temperature ${T}_{c}=\frac{{q}^{2}}{2{k}_{B}}$ through the use of upper and lower bounds (valid for all $T>{T}_{c}$) for the canonical partition function ${Q}^{*}$ with lim ${Q}^{*}\ensuremath{\sim}N!{({Q}_{1}^{*})}^{N}$ as $T\ensuremath{\rightarrow}{T}_{c}^{+}$, ${Q}_{1}^{*}$ denoting the restriction of ${Q}^{*}$ to a pair (+ -). Below ${T}_{c}$, the equilibrium properties are investigated with the use of the binary approximation proposed by Hauge and Hemmer for charged disks. The resolution of the two-body Schr\"odinger equation allows us to consider point particles and place on a firm basis preliminary conclusions about the divergent behavior of the thermodynamic functions. The pair-correlation function ${g}_{2}(r)$ is investigated above ${T}_{c}$ for the one-component model within the framework of the Debye approximation, through a potential of average force ${w}_{2}(r)$, up to the third order in the plasma parameter $\frac{{q}^{2}}{{k}_{B}}T$. The short-range behavior of ${w}_{2}(r)$ appears as a renormalizable quantity, while the long-range behavior confirms and extends three-dimensional findings. The $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the corresponding thermodynamic functions coincides with exact results derived by Mehta for the same model restricted to the unit circumference. Finally, the Debye free energy is shown to fulfill a sufficient condition required by the existence of the thermodynamic limit.