Factorial moments of continuous order

Abstract

The normalized factorial moments F q are continued to noninteger values of the order q, satisfying the condition that the statistical fluctuations remain filtered out. That is, for Poisson distribution F q = 1 for all q. The continuation procedure is designed with phenomenology and data analysis in mind. Examples are given to show how F q can be obtained for positive and negative values of q. With q being continuous, multifractal analysis is made possible for multiplicity distributions that arise from self-similar dynamics. A step-by-step procedure of the method is summarized in the conclusion.

Chaotic behavior of particle production in branching processes

“…Their drawback is that the statistical fluctuations must be taken out of G q explicitly by hand. More recently, a method for continuing F q to noninteger q, while maintaining its virtue of not being contaminated by statistical fluctuations, has been developed [18]. We shall therefore make use of it to determine D 1 .…”

“…Using P n as input from experiment or from simulation, one can determine f q , but only for integer q. One may replace the factorials in (18) by gamma functions in order to continue (18) to noninteger q [20]. But that procedure does not result in F q = 1 for Poisson distribution for all q.…”

In Search for Signs of Chaos in Branching Processes

“…To that end, it is necessary to extend the definition of Fq in (6) to noninteger q. A method for achieving that without losing the attribute Fq = 1 for Poissonian fluctuation has recently been developed [14]. Using that method, we have calculated Fq for a continuous range of q, as shown in Fig.…”

Multiplicity distributions of squeezed isospin states

“…The other is the lack of a theoretical framework to describe the unusual charge distributions independent of the chiral dynamics. It seems to us useful to have a language to convey quantitatively the characteristics of the distributions without a rigid dynamical basis so that one may have what amounts to a representation in much the same way that the negative binomial distributions have been used to describe multiplicity distributions in general [4,18,19]. While there exist earlier attempts to describe exotic neutral and charge distributions in terms of the coherent states [2,3], it is better for our purpose to introduce some parameters that can be adjusted to change the neutral-to-charge ratio.…”