Inverse problem for Planck formula

TL;DR

This paper addresses the inverse problem of deriving the energy distribution function of black body radiation from the Planck formula, considering ensembles with positive and negative temperatures and resolving related paradoxes.

Contribution

It provides a solution to the inverse problem for Gibbs ensembles with both positive and negative temperatures, expanding the understanding of black body radiation distributions.

Findings

Solution to the inverse problem for ensembles with positive and negative temperatures

Analysis of ensembles with finite energies and phase volumes

Discussion of the absence of Bohr-van Leeuwen paradox in these ensembles

Abstract

Planck formula is considered as a first moment (average value) of unknown function of electromagnetic energy distribution of black body radiation. In-verse problem for the definition of the unknown function is solved for Gibbs ensemble. The solution needs of ensembles with both absolute temperatures: positive temperature and negative temperature. Such ensembles are the part of more extended class of ensembles with finite energies and finite phase vol-umes. In addition, the absence of Bohr - van Leeuwen paradox is considered for such statistical ensembles.