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To extend the Bose-Einstein (BE) distribution to fractional order, we turnour attention to the differential equation, $df/dx =-f-f^2$. It is satisfiedwith the stationary solution, $f(x)=1/(e^{x+\mu}-1)$, of the Kompaneetsequation, where $\mu$ is the constant chemical potential. Setting $R=1/f$, weobtain a linear differential equation for $R$. Then, the Caputo fractionalderivative of order $p$ ($p>0$) is introduced in place of the derivative of$x$, and fractional BE distribution is obtained, where function ${\rm e}^x$ isreplaced by the Mittag-Leffler (ML) function $E_p(x^p)$. Using the integralrepresentation of the ML function, we obtain a new formula. Based on theanalysis of the NASA COBE monopole data, an identity $p\simeq e^{-\mu}$ isfound.

journal of physics. conference series

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