  Various phenomenological models of particle multiplicity distributions are
discussed using a general form of the grand canonical partition function. These
phenomenological models include a wide range of varied processes such as
coherent emission or Poisson processes, chaotic emission resulting in a
negative binomial distribution, combinations of coherent and chaotic processes
called signal/noise distributions, and models based on field emission from
Lorentzian line shapes leading to Lorentz/Catalan distributions. These specific
cases can be written as special cases of a more general distribution. Using
this grand canonical approach moments and cumulants, combinants, hierarchical
structure, void scaling relations, KNO scaling features, clan variables and
branching laws associated with stochastic or ancestral variables are discussed.
It is shown that just looking at the mean and fluctuation of data is not enough
to distinguish these distributions or the underlying mechanism. A
generalization of the Poisson transform of a distribution and the Poissonian
decomposition of it into a compound or sequential process is also given.
