  We present a calculation of the grand canonical partition function of a
serial metallic island system by the imaginary-time path integral formalism. To
this purpose, all electronic excitations in the lead and island electrodes are
described using Grassmann numbers. Coulomb charging energy of the system is
represented in terms of phase fields conjugate to the island charges. By the
large channel approximation, the tunneling action phase dependence can also be
determined explicitly. Therefore, we represent the partition function as a path
integral over phase fields with a path probability given in an analytically
known effective action functional. Using the result, we also propose a
calculation of the average electron number of the serial island system in terms
of the expectation value of winding numbers. Finally, as an example, we
describe the Coulomb blockade effect in the two-island system by the average
electron number and propose a method to construct the quantum stability
diagram.
