Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids

In a Coulomb fluid confined to a domain $V$, the dielectric susceptibility tensor $\chi_V$ depends on the shape of $V$, even in the thermodynamic $V\to\infty$ limit. This paper deals with the classical two-dimensional one-component plasma formulated in an elliptic $V$-domain, equilibrium statistical mechanics is used. For the dimensionless coupling constant $\Gamma =$ even positive integer, the mapping of the plasma onto a discrete one-dimensional anticommuting-field theory provides a new sum rule. This sum rule confirms the limiting value of $\chi_V$ predicted by macroscopic electrostatics and gives the finite-size correction term to $\chi_V$.