  We compare ergodic properties of the kinetic energy for three stochastic
models of subrecoil-laser-cooled gases. One model is based on a heterogeneous
random walk (HRW), another is an HRW with long-range jumps (the exponential
model), and the other is a mean-field-like approximation of the exponential
model (the deterministic model). All the models show an accumulation of the
momentum at zero in the long-time limit, and a formal steady state cannot be
normalized, i.e., there exists an infinite invariant density. We obtain the
exact form of the infinite invariant density and the scaling function for the
exponential and deterministic models and devise a useful approximation for the
momentum distribution in the HRW model. While the models are kinetically
non-identical, it is natural to wonder whether their ergodic properties share
common traits, given that they are all described by an infinite invariant
density. We show that the answer to this question depends on the type of
observable under study. If the observable is integrable, the ergodic properties
such as the statistical behavior of the time averages are universal as they are
described by the Darling-Kac theorem. In contrast, for non-integrable
observables, the models in general exhibit non-identical statistical laws. This
implies that focusing on non-integrable observables, we discover non-universal
features of the cooling process, that hopefully can lead to a better
understanding of the particular model most suitable for a statistical
description of the process. This result is expected to hold true for many other
systems, beyond laser cooling.
