Fractional Fokker-Planck Equation in Time Variable and Oscillation of Cumulant Moments

Abstract

Fractional derivative in time variable is introduced into the Fokker-Planck equation of a population growth model. It's solution, the KNO scaling function, is transformed into the generating function for the multiplicity distribution. Formulas of the factorial moment and the $H_j$ moment are derived from the generating function, which reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. In our approach, oscillation of $H_j$ moment appears contrary to the case of the NBD. Calculated $H_j$ moments are compared with those given from the data in $p\bar{p}$ collisions and in $e^+e^-$ collisions.