  The analytic asymptotic expressions for the Casimir free energy, pressure and
entropy at low temperature in the configuration of one metal and one dielectric
plate are obtained. For this purpose we develop the perturbation theory in a
small parameter proportional to the product of the separation between the
plates and the temperature. This is done using both the simplified model of an
ideal metal and of a dielectric with constant dielectric permittivity and for
the realistic case of the metal and dielectric with frequency-dependent
dielectric permittivities. The analytic expressions for all related physical
quantities at high temperature are also provided. The obtained analytic results
are compared with numerical computations and good agreement is found. We
demonstrate for the first time that the Lifshitz theory, when applied to the
configuration of metal-dielectric, satisfies the requirements of thermodynamics
if the static dielectric permittivity of a dielectric plate is finite. If it is
infinitely large, the Lifshitz formula is shown to violate the Nernst heat
theorem. The implications of these results for the thermal quantum field theory
in Matsubara formulation and for the recent measurements of the Casimir force
between metal and semiconductor surfaces are discussed.
