Fractional Fokker-Planck equation and oscillatory behavior of cumulant moments.

The Fokker-Planck equation is considered, which is connected to the birth and death process with immigration by the Poisson transform. The fractional derivative in time variable is introduced into the Fokker-Planck equation in order to investigate an origin of oscillatory behavior of cumulant moments. From its solution (the probability density function), the generating function (GF) for the corresponding probability distribution is derived. We consider the case when the GF reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. The H(j) moment derived from the GF of the NBD decreases monotonically as the rank j increases. However, the H(j) moment derived in our approach oscillates, which is contrasted with the case of the NBD. Calculated H(j) moments are compared with those of charged multiplicities observed in pp, e(+)e(-), and e(+)p collisions. A phenomenological meaning of introducing the fractional derivative in time variable is discussed.