The finite-temperature Casimir free energy, entropy, and internal energy are considered anew for a conventional parallel-plate configuration, in the light of current discussions in the literature. In the case of an "ideal" metal, characterized by a refractive index equal to infinity for all frequencies, we recover, via a somewhat unconventional method, conventional results for the temperature dependence, meaning that the zero-frequency transverse electric mode contributes the same as the transverse magnetic mode. For a real metal, however, approximately obeying the Drude dispersive model at low frequencies, we find that the zero-frequency transverse electric mode does not contribute at all. This would appear to lead to an observable temperature dependence and a violation of the third law of thermodynamics. It had been suggested that the source of the difficulty was the behavior of the reflection coefficient for perpendicular polarization but we show that this is not the case. By introducing a simplified model for the Casimir interaction, consisting of two harmonic oscillators interacting via a third one, we illustrate the behavior of the transverse electric field. Numerical results are presented based on the refractive index for gold. A linear temperature correction to the Casimir force between parallel plates is indeed found which should be observable in room-temperature experiments, but this does not entail any thermodynamic inconsistency.

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The temperature dependence of the Casimir force between a real metallic plate and a metallic sphere is analyzed on the basis of optical data concerning the dispersion relation of metals such as gold and copper. Realistic permittivities imply, together with basic thermodynamic considerations, that the transverse electric zero mode does not contribute. This results in observable differences with the conventional prediction, which does not take this physical requirement into account. The results are shown to be consistent with the third law of thermodynamics, as well as being not inconsistent with current experiments. However, the predicted temperature dependence should be detectable in future experiments. The inadequacies of approaches based on ad hoc assumptions, such as the plasma dispersion relation and the use of surface impedance without transverse momentum dependence, are discussed.

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near T = 0 we find the leading term proportional to T 2 and the next-to-leading term proportional to T 5/2 . These terms give rise to zero Casimir entropy as T → 0, and is thus in accordance with Nernst's theorem.

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