Generalizing the Planck distribution

TL;DR

This paper proposes a heuristic generalization of Planck's black-body radiation law using nonextensive statistical mechanics and superstatistics, leading to a simple differential equation that recovers the original law as a special case.

Contribution

It introduces a novel, mathematically elegant generalization of Planck's law based on nonextensive entropy and superstatistics, with potential applications in complex systems.

Findings

Generalized Planck law via differential equation approach

Recovers original law as a special case when q=2

Provides a framework for exploring out-of-equilibrium phenomena

Abstract

Along the lines of nonextensive statistical mechanics, based on the entropy $S_q = k(1- \sum_i p_i^q)/(q-1) (S_1=-k \sum_i p_i \ln p_i)$, and Beck-Cohen superstatistics, we heuristically generalize Planck's statistical law for the black-body radiation. The procedure is based on the discussion of the differential equation $dy/dx=-a_{1}y-(a_{q}-a_{1}) y^{q}$ (with $y(0)=1$), whose $q=2$ particular case leads to the celebrated law, as originally shown by Planck himself in his October 1900 paper. Although the present generalization is mathematically simple and elegant, we have unfortunately no physical application of it at the present moment. It opens nevertheless the door to a type of approach that might be of some interest in more complex, possibly out-of-equilibrium, phenomena.