  This paper presents a new derivation of the Generalized Poisson distribution.
This distribution provides a good fit to the evolved, counts-in-cells
distribution measured in numerical simulations of hierarchical clustering from
Poisson initial conditions. This new derivation can be used to construct a
useful analytic model of the evolution of clustering measured in these
simulations. The model is consistent with the assumption that, as the universe
expands and the comoving sizes of regions change as a result of gravitational
instability, the number of such expanding and contracting regions is conserved.
The model neglects the influence of external tides on the evolution of such
regions. Indeed, in the context of this model, the GPD can be thought of as
arising from a simple variant of the well-studied spherical collapse model, in
which tidal effects are also neglected. The model can be understood as a simple
generalization of the excursion set model which is used to estimate the number
density of collapsed, virialized halos. The generalization allows one to
describe the spatial distribution of the halos and the mass simultaneously, so
the extent to which these halos are biased tracers of the underlying matter
distribution is easy to estimate.
