Dispersion forces between real metal plates at finite temperature and the Nernst heat theorem

We derive the zero-and-finite-temperature dispersion-forces in terms ofchanges in the zero-point energy of the electromagnetic normal modes of thesystem. At finite temperature the Helmholtz free energy of interactiondetermines the force. We express this energy in two complementary ways; inthe first as a discrete summation over complex frequencies, Matsubarafrequencies; in the second as an integral along the real frequency axis. We use these expressions to get the entropy of interaction and demonstratethat this entropy obeys the Nernst heat theorem, one version of the thirdlaw of thermodynamics