Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids II

Abstract

For a Coulomb system contained in a domain \Lambda, the dielectric susceptibility tensor \chi_{\Lambda} is defined as relating the average polarization in the system to a constant applied electric field, in the linear limit. According to the phenomenological laws of macroscopic electrostatics, \chi_{\Lambda} depends on the specific shape of the domain \Lambda. In this paper we derive, using the methods of equilibrium statistical mechanics in both canonical and grand-canonical ensembles, the shape dependence of \chi_{\Lambda} and the corresponding finite-size corrections to the thermodynamic limit, for a class of general \nu-dimensional (\nu\ge 2) Coulomb systems, of ellipsoidal shape, being in the conducting state. The microscopic derivation is based on a general principle: the total force acting on a system in thermal equilibrium is zero. The results are checked in the Debye-H\"uckel limit. The paper is a generalization of a previous one [L. \v{S}amaj, J. Stat. Phys. 100:949 (2000)], dealing with the special case of a one-component plasma in two dimensions. In that case, the validity of the presented formalism has already been verified at the exactly solvable (dimensionless) coupling \Gamma = 2.