Thermodynamics of the Bose Gas and Blackbody Radiation in Non-Extensive Tsallis Statistics
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modynamics of the Bose Gas and Blackbody Radiation in Non-Extensive Tsallis Statistics A. V. Kolesnichenko* Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia *e-mail: [email protected] Received February 26, 2020; revised April 20, 2020; accepted May 25, 2020
Abstract—The aim of this paper is to construct the thermodynamics of open quantum systems of elementary Bose-gas particles in the framework of nonextensive Tsallis statistics based on modified quantum entropy, which depends on the real deformation parameter q. We obtain generalized expressions for the thermodynamic potential, internal energy, free energy, specific heat, and pressure, as well as the basic thermodynamic equations. Modified equilibrium Bose–Einstein distributions for massive particles and generalized Planck, Rayleigh–Jeans, and Wien laws for photons that can be applied to various physical problems, in particular, to the description of cosmic blackbody radiation, are discussed. The initial basis for such a consideration of a photon gas is the assumption that the photon distribution of the cosmic background radiation (in thermal equilibrium) may differ from the classical Planck distribution due to the influence of long-range gravitational effects at large distances. This influence is probably a reflection of the fact that in remote time matter and light were strongly interconnected. The generalized thermodynamics of a photon gas can be used, in particular, as a theoretical justification for experimental studies of blackbody radiation inside various astrophysical objects. Keywords: Tsallis statistical mechanics, nonextensive Bose-gas entropy, blackbody radiation, Bregman divergence DOI: 10.1134/S0038094620050020
INTRODUCTION As it has now become clear, the Boltzmann–Gibbs (BG) statistical mechanics and standard thermodynamics are not completely universal theories, since they have limited applicability domains. An example is the impossibility, within the framework of BG statistics, of explaining the spectrum of cosmic rays, one of the most important systems of relativistic particles. Numerous examples of complex (anomalous) systems that have strong long-range effects, nonlocal correlations between individual elements of a system (remembering its past), the fractal nature of phase space, and non-Markov behavior are known in physics and other natural sciences using the methods of statistical mechanics. The complex space-time structure of such systems leads to a violation of the additivity principle for such important thermodynamic characteristics as entropy or internal energy. In this regard, research in the field of mechanics of nonextensive (nonadditive) systems has recently become a subject of considerable interest, due to both the novelty of the general theoretical problems that arise here and the importance of practical applications. The beginning of a systematic study in this direction is connected with the work of Tsallis (Tsallis,
1988), in which the entropy functional was introduced −1 q Sq ( p) := k(
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