  So far there has been no reliable method to calculate the Casimir force at
separations comparable to the root-mean-square of the height fluctuations of
the surfaces. Statistical analysis of rough gold samples has revealed the
presence of peaks considerably higher than the root-mean-square roughness.
These peaks redefine the minimum separation distance between the bodies and can
be described by extreme value statistics. Here we show that the contribution of
the high peaks to the Casimir force can be calculated with a pairwise additive
summation, while the contribution of asperities with normal height can be
evaluated perturbatively. This method provides a reliable estimate of the
Casimir force at short distances, and it solves the significant, so far
unexplained discrepancy between measurements of the Casimir force between rough
surfaces and the results of perturbation theory. Furthermore, we illustrate the
importance of our results in a technologically relevant situation.
