  We develop a theory for the analytic computation of the free energy of band
insulators in the presence of a uniform and constant electric field. The two
key ingredients are a perturbation-like expression of the Wannier-Stark energy
spectrum of electrons and a modified statistical mechanics approach involving a
local chemical potential in order to deal with the unbounded spectrum and
impose the physically relevant electronic filling. At first order in the field,
we recover the result of King-Smith, Vanderbilt and Resta for the electric
polarization in terms of a Zak phase - albeit at finite temperature - and, at
second order, deduce a general formula for the electric susceptibility, or
equivalently for the dielectric constant. Advantages of our method are the
validity of the formalism both at zero and finite temperature and the easy
computation of higher order derivatives of the free energy. We verify our
findings on two different one-dimensional tight-binding models.

Strongly Correlated Electrons;Statistical Mechanics