Statistical mechanics of dipolar fluids: dielectric constant and sample shape

We give a new proof that the constitutive relation of macroscopic electrostatics holds in a dipolar fluid with a sample shape independent dielectric constant. Our approach is based on a BGY-like hierarchy equation which allows us to calculate the canonical one-body density function up to linear order in the electric field, in the thermodynamic limit. The dielectric constant comes out as an integral over a 3-point correlation function of the infinite unperturbed (unpolarized) system, from which one can recover the well-known formula for in terms of the 2-point direct correlation function.