Fokker-Planck stochastic Koba-Nielsen-Olesen solutions, branching processes, and path to hadronization

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Abstract
We have analyzed a large class of Markov processes for the properties of multiplicity distribution at finite n¯ and for their Koba-Nielsen-Olesen (KNO) scaling and scaling violation by using a Fokker-Planck stochastic differential equation of the KNO-scaling functions, QCD branching processes, and/or simple supercluster models. Through this analysis, the distinction between the simple Fokker-Planck formulation of the KNO function and the general branching processes is seen. Our solutions allow in general an arbitrary initial condition and demonstrate the richness and flexibility of many possible paths of hadronization. They can be used for the study of distribution of the preexisting gluons and quarks.