Sine-Gordon theory for the equation of state of classical hard-core Coulomb systems. I Low fugacity expansion

We present an exact field theoretical representation of the statistical mechanics of classical hard-core Coulomb systems. This approach generalizes the usual sine-Gordon theory valid for point-like charges or lattice systems to continuous Coulomb fluids with additional short-range interactions. This formalism is applied to derive the equation of state of the restricted primitive model of electrolytes in the low fugacity regime up to order $\rho^{5/2}$ ($\rho$ number density). We recover the results obtained by Haga by means of Mayer graphs expansions.