Electrostatic Interaction due to Patch Potentials on Smooth Conducting Surfaces

We evaluate the electrostatic interaction energy between two surfaces, one flat and the other slightly curved, in terms of the two-point autocorrelation functions for patch potentials on each one of them, and of a single function $\ensuremath{\psi}$ which defines the curved surface. The resulting interaction energy, a functional of $\ensuremath{\psi}$, is evaluated up to the second order in a derivative expansion approach. We derive explicit formulas for the coefficients of that expansion as simple integrals involving the autocorrelation functions, and evaluate them for some relevant patch-potential profiles and geometries.