Polylogarithmic Representation of Radiative and Thermodynamic Properties of Thermal Radiation in a Given Spectral Range: I. Blackbody Radiation

Abstract

The thermodynamics of blackbody radiation has been constructed for the entire range of the spectrum. However, in practical applications, thermodynamic functions must be calculated within a finite range of frequencies. The analytical expressions for the radiative and thermodynamic properties of blackbody radiation over an arbitrary spectral range of the electromagnetic spectrum are obtained. The Wien displacement law, Stefan–Boltzmann law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, and pressure are expressed in terms of the polylogarithm functions. These expressions are important when we build a theoretical model of radiative heat transfer, for example. The thermodynamic functions of blackbody radiation are calculated for various ranges of the spectrum at different temperatures. As an example of practical applications, thermodynamics of the cosmic microwave background radiation measured by the COBE FIRAS instrument is constructed. The expressions obtained for the radiative and thermodynamic functions of blackbody radiation can easily be presented in wavelength and wavenumber domains.