2021-10-26T16:35:41Zhttps://soar-ir.repo.nii.ac.jp/oaioai:soar-ir.repo.nii.ac.jp:000199942021-09-02T06:28:18Z1169:1170An analytic relation between the fractional parameter in the Mittag–Leffler function and the chemical potential in the Bose–Einstein distribution through the analysis of the NASA COBE monopole dataBiyajima, MinoruMizoguchi, TakuyaSuzuki, NaomichiTo extend the Bose-Einstein (BE) distribution to fractional order, we turn our attention to the differential equation, df/dx = −f − f 2. It is satisfied with the stationary solution, f(x) = 1/(e x + μ − 1), of the Kompaneets equation, where μ is the constant chemical potential. Setting R = 1/f, we obtain a linear differential equation for R. Then, the Caputo fractional derivative of order p (p > 0) is introduced in place of the derivative of x, and fractional BE distribution is obtained, where function e x is replaced by the Mittag–Leffler (ML) function Ep(x p ). Using the integral representation of the ML function, we obtain a new formula. Based on the analysis of the NASA COBE monopole data, an identity p sime e −μ is found.ArticleJournal of Physics: Conference Series. 936(1):12082(2018)journal articleIOP Publishing Ltd2018-02-19application/pdfJournal of Physics: Conference Series1936120821742-6588https://soar-ir.repo.nii.ac.jp/record/19994/files/Biyajima_2017_J._Phys.%3A_Conf._Ser._936_012082.pdfeng10.1088/1742-6596/936/1/012082https://doi.org/10.1088/1742-6596/936/1/012082Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd