An extension of the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions by the Caputo fractional derivative

Minoru Biyajima, Takuya Mizoguchi, Naomichi Suzuki

Published 2015-04-06Version 1

In order to generalize the Maxwell-Boltzmann (MB), Bose-Einstein (BE), and Fermi-Dirac (FD) distributions to fractional order, we start with the thermodynamical equation, $\partial U/\partial \beta =-aU-bU^2$, with $\beta =1/k_BT$ and parameters $a$ ($a>0$) and $b$, which is equivalent to the equation proposed by Planck in 1900. Setting $R=1/U$ and $x=a(\beta-\beta_0)$, we obtain the linear partial differential equation $\partial R/\partial x = R + b/a$ from the thermodynamical equation. Then, the Caputo fractional derivative of order $p$ ($p>0$) is introduced in place of the partial derivative of $x$. We obtain the fractional MB, BE, and FD distributions, where the exponential function, ${\rm e}^x$, is replaced by the Mittag-Leffler function, $E_p(x^p)$. The behaviors of the fractional FD distribution are examined.