The effects of the chemical potential in a BE distribution and the fractional parameter in a distribution with Mittag-Leffler function

TL;DR

This paper derives a fractional Bose-Einstein distribution using Caputo derivatives and Mittag-Leffler functions, analyzing NASA COBE data to relate the fractional parameter to chemical potential.

Contribution

It introduces a novel fractional distribution model with Mittag-Leffler functions and links the fractional parameter to chemical potential based on observational data.

Findings

Identifies a relationship p ≈ e^{−μ} between fractional parameter and chemical potential.

Provides a new formula for fractional BE distribution using Mittag-Leffler functions.

Analyzes COBE data to support the theoretical model.

Abstract

The fractional Planck distribution is calculated by applying the Caputo fractional derivative with order $p$ ($p > 0$) to the equation proposed by Planck in 1900. In addition, the integral representation of the Mittag--Leffler function is employed to obtain a new formula for the fractional BE distribution, which is then used to analyze the NASA COBE monopole data. Based on this analysis, an identity $p\simeq e^{-\mu}$ is found, where $\mu$ is the dimensionless constant chemical potential that was introduced to the BE distribution by the NASA COBE collaboration.