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In this paper, we consider the nonequilibrium Casimir pressure in the system of two parallel graphene-coated plates one of which is either warmer or cooler than the environment. The electromagnetic response of graphene coating characterized by the nonzero energy gap and chemical potential is described in the framework of the Dirac model by means of the polarization tensor. It is shown that the magnitude of the nonequilibrium Casimir pressure on a warmer plate than the environment is larger and on a cooler plate it is smaller than the magnitude of the standard Casimir pressure in the state of thermal equilibrium. According to our results, the spatially local theory underestimates the role of the effects of nonequilibrium. This underestimation increases for a smaller chemical potential of the graphene coating and at lower temperatures of the cooled plate. Possible applications of the obtained results are indicated.

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature $T$ as $q=\sigma T^4$, with $\sigma =5.67\times 10^{-8}$Wm${}^{-2}$K${}^{-4}$ the SB constant, first estimated by Stefan to within $11\%$ of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation implies two assumptions not justified within a classical approach, namely: (i) the zero value of the chemical potential and (ii) the internal energy of the black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant $h$. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.

The Casimir pressure on the dielectric layer in metal-dielectric-semiconductor (MDS) structures is calculated in the framework of the Lifshitz theory at nonzero temperature. In this calculation the standard parameters of semiconductor devices with a thin dielectric layer are used. We consider the thickness of a layer decreasing from 40 to 1 nm. At the shortest thickness the Casimir pressure achieves 8 MPa. At small thicknesses the results are compared with the predictions of nonrelativistic theory.