  Various models for pion multiplicity distributions produced in relativistic
heavy ion collisions are discussed. The models include a relativistic
hydrodynamic model, a thermodynamic description, an emitting source pion laser
model, and a description which generates a negative binomial description. The
approach developed can be used to discuss other cases which will be mentioned.
The pion probability distributions for these various cases are compared.
Comparison of the pion laser model and Bose-Einstein condensation in a laser
trap and with the thermal model are made. The thermal model and hydrodynamic
model are also used to illustrate why the number of pions never diverges and
why the Bose-Einstein correction effects are relatively small. The pion
emission strength $\eta$ of a Poisson emitter and a critical density $\eta_c$
are connected in a thermal model by $\eta/n_c = e^{-m/T} < 1$, and this fact
reduces any Bose-Einstein correction effects in the number and number
fluctuation of pions. Fluctuations can be much larger than Poisson in the pion
laser model and for a negative binomial description. The clan representation of
the negative binomial distribution due to Van Hove and Giovannini is discussed
using the present description. Applications to CERN/NA44 and CERN/NA49 data are
discussed in terms of the relativistic hydrodynamic model.
