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

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.