The Casimir force and free energy at low temperatures has been the subject of focus for some time. We calculate the temperature correction to the Casimir-Lifshitz free energy between two parallel plates made of dielectric material possessing a consta
nt conductivity at low temperatures, described through a Drude-type dielectric function. For the transverse magnetic (TM) mode such a calculation is new. A further calculation for the case of the TE mode is thereafter presented which extends and generalizes previous work for metals. A numerical study is undertaken to verify the correctness of the analytic results.
In this Reply to the preceding Comment of Klimchitskaya and Mostepanenko (cf. also quant-ph/0703214), we summarize and maintain our position that the Drude dispersion relation when inserted in the Lifshitz formula gives a thermodynamically satisfacto
ry description of the Casimir force, also in the limiting case when the relaxation frequency goes to zero (perfect crystals).
In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersive model, assuming the relaxation is nonzero at zero temperature (which is the case when impurities ar
e present), gives consistent results for the Casimir free energy at low temperatures. Specifically, we find that the free energy consists essentially of two terms, one leading term proportional to T^2, and a next term proportional to T^{5/2}. Both these terms give rise to zero Casimir entropy as T -> 0, thus in accordance with Nernsts theorem.
