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 of changes in the zero-point energy of the electromagnetic normal modes of the system. At finite temperature the Helmholtz free energy of interaction determines the force. We express this energy in two complementary ways; in the first as a discrete summation over complex frequencies, Matsubara frequencies; in the second as an integral along the real frequency axis. We use these expressions to get the entropy of interaction and demonstrate that this entropy obeys the Nernst heat theorem, one version of the third law of thermodynamics.