Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids II

Details Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids II Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids II

2003-08-15

arxiv.org/abs/cond-mat/0308304v1

J. Stat. Phys. 114 (2004) 1211-1234

Physics

Condensed Matter

Statistical Mechanics

22 pages

Scientific paper

10.1023/B:JOSS.0000013972.61656.

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.

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