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An 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 data
Biyajima, Minoru
Mizoguchi, Takuya
Suzuki, Naomichi
Content 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
To 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.
Article
Journal of Physics: Conference Series. 936(1):12082(2018)
IOP Publishing Ltd
2018-02-19
eng
journal article
AM
http://hdl.handle.net/10091/00020755
https://soar-ir.repo.nii.ac.jp/records/19994
https://doi.org/10.1088/1742-6596/936/1/012082
10.1088/1742-6596/936/1/012082
1742-6588
Journal of Physics: Conference Series
936
1
12082
https://soar-ir.repo.nii.ac.jp/record/19994/files/Biyajima_2017_J._Phys.%3A_Conf._Ser._936_012082.pdf
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2018-08-08