Polylogarithmic representation of radiative and thermodynamic properties of thermal radiation in a given spectral range: I. Blackbody radiation

TL;DR

This paper derives exact analytical formulas for blackbody radiation properties within finite spectral ranges using polylogarithm functions, enabling practical calculations for applications like the Cosmic Microwave Background.

Contribution

It introduces new polylogarithm-based expressions for blackbody radiative and thermodynamic properties in finite frequency ranges, enhancing computational and practical utility.

Findings

Derived explicit formulas for various thermodynamic properties.

Applied formulas to CMB data from COBE FIRAS.

Facilitated calculations in wavelength and wavenumber domains.

Abstract

Using polylogarithm functions the exact analytical expressions for the radiative and thermodynamic properties of blackbody radiation, such as 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 in the finite range of frequencies are constructed. The obtained expressions allow us to tabulate these functions in various finite frequency bands at different temperatures for practical applications. As an example, the radiative and thermodynamic functions using experimental data for the monopole spectrum of the Cosmic Microwave Background (CMB) radiation measured by the COBE FIRAS instrument in the 60 - 600 GHz frequency interval at the temperature T = 2.725 K are calculated. The expressions obtained for the radiative and thermodynamic functions can be easily presented in wavelength and wavenumber domains.