The low-temperature asymptotic expressions for the Casimir interaction between two real metals described by Leontovich surface impedance are obtained in the framework of thermal quantum field theory. It is shown that the Casimir entropy computed using the impedance of infrared optics vanishes in the limit of zero temperature. By contrast, the Casimir entropy computed using the impedance of the Drude model attains at zero temperature a positive value which depends on the parameters of a system, i.e., the Nernst heat theorem is violated. Thus, the impedance of infrared optics withstands the thermodynamic test, whereas the impedance of the Drude model does not. We also perform a phenomenological analysis of the thermal Casimir force and of the radiative heat transfer through a vacuum gap between real metal plates. The characterization of a metal by means of the Leontovich impedance of the Drude model is shown to be inconsistent with experiment at separations of a few hundred nanometers. A modification of the impedance of infrared optics is suggested taking into account relaxation processes. The power of radiative heat transfer predicted from this impedance is several times less than previous predictions due to different contributions from the transverse electric evanescent waves. The physical meaning of low frequencies in the Lifshitz formula is discussed. It is concluded that new measurements of radiative heat transfer are required to find out the adequate description of a metal in the theory of electromagnetic fluctuations.
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