Thermodynamic Properties of the Two-Dimensional Coulomb Gas in the Low-Density Limit

Abstract

The model under consideration is the two-dimensional Coulomb gas of $\pm$ charged hard disks with diameter $σ$. For the case of pointlike charges $(σ=0)$, the system is stable against collapse of positive-negative pairs of charges in the range of inverse temperatures $0 \le β< 2$, where its full thermodynamics was obtained exactly [L. {Š}amaj and I. Trav{ě}nec, {\it J. Stat. Phys.} {\bf 101}:713 (2000)]. In the present work, we derive the leading correction to the exact thermodynamics of pointlike charges due to the presence of the hard core $σ$ (appearing in the dimensionless combination $nσ^2$, $n$ is the particle density). This permits us to extend the treatment to the interval $2\le β<3$ (the Kosterlitz-Thouless phase transition takes place at $β=4$). The results, which are exact in the low-density limit $nσ^2 \to 0$, reproduce correctly the singularities of thermodynamic quantities at the collapse point $β=2$ and agree very well with Monte-Carlo simulations.