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

Minoru Biyajima,Naomichi Suzuki,Takuya Mizoguchi

doi:10.1088/1742-6596/936/1/012082

Abstract

To extend the Bose-Einstein (BE) distribution to fractional order, we turn our attention to the differential equation, df/dx = −f − f2. It is satisfied with the stationary solution, f(x) = 1/(ex + 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 ex is replaced by the Mittag–Leffler (ML) function Ep(xp). 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 ≃ e−μ is found.

Highlights

The COBE FIRAS experiments have shown that the cosmic microwave background (CMB) radiation spectrum is well described by the Planck distribution with temperature, T = 2725.0±1 mK [1, 2]
In Appendix A, the Caputo fractional derivative is introduced into Eq (5) in place of the derivative x, and a fractional BE and other distributions are obtained
In [11], we have applied the Riemann–Liouville fractional derivative to obtain a fractional BE distribution f (x) = 1/(Ep(xp) − 1), and we have investigated the NASA COBE monopole data using BE and fractional BE distributions

Summary

Introduction

Equation (2) describes the photon distribution, which obeys the Planck distribution at the initial stage, and is affected by the elastic e-γ scatterings in the expanding universe. In Appendix A, the Caputo fractional derivative is introduced into Eq (5) in place of the derivative x, and a fractional BE and other distributions are obtained.

Results

Conclusion