Coulomb Systems Seen as Critical Systems: Ideal Conductor Boundaries

The grand potential of a classical Coulomb system has universal finite-size corrections similar to the ones which occur in the free energy of a simple critical system : the massless Gaussian field. Here, the Coulomb system is assumed to be confined by walls made of an ideal conductor material; this choice corresponds to simple (Dirichlet) boundary conditions for the Gaussian field. For a $d$-dimensional ($d>or=2$) Coulomb system confined in a slab of thickness $W$, the grand potential (in units of $kT$) per unit area has the universal term $Gamma(d/2) zeta(d)/2^d pi^{d/2}W^{d-1}$. For a two-dimensional Coulomb system confined in a disk of radius $R$, the grand potential (in units of $kT$) has the universal term $(1/6) ln R$. These results, of general validity, are checked on two-dimensional solvable models.