We present a complete derivation of the granular analogue to Drude
conductivity using diagrammatic methods. The convergence issues arising when
changing the order of momentum and frequency summation are more severe than in
the homogeneous case. This is because there are now two momentum sums rather
than one, due to the intragrain momentum scrambling in tunnelling events. By
careful analytic continuation of the frequency sum, and use of integration by
parts, we prove that the system is in the normal (non-superconducting) state,
and derive the formula for the granular Drude conductivity expected from
Einstein's relation and Fermi's golden rule. We also show that naively
performing the momentum sums first gives the correct result, provided that we
interpret a divergent frequency sum by analytic continuation using the Hurwitz
zeta function.Comment: 18 pages, 5 figure
