Fluctuation theorems for dielectrics with periodic boundary conditions

We derive fluctuation theorems for dielectrics with periodic boundary conditions defined by a Bravais lattice in which the configuration of a large number of permanent dipoles in the unit cell is repeated periodically. We use the electrostatic approximation and show that it is essential to consider bounded geometry. We consider in particular geometries adapted to the Ewald summation used in computer calculations, namely ellipsoids, with as special cases a slab and a sphere. The fluctuations depend strongly on the chosen geometry.