  (Abridged) We revisit the scaling relationships between the dark matter mass
and observed X-ray luminosity and temperature of galaxy clusters and groups in
the local Universe. Specifically, we compare recent observations with analytic
models of the intracluster medium in which the gas entropy distribution has
been shifted by a variable amount, K0, to investigate the origin of the scatter
in these scaling relations, and its influence on the luminosity and temperature
functions. We find that variations in halo concentration or formation epoch
(which might determine the time available for low entropy gas to cool out) are
insufficient to explain the amount of scatter in the mass-luminosity relation.
Instead, a range of entropy floors at a fixed halo mass, spanning approximately
50 keV sq.cm to 700 keV sq.cm, is required to match the data. This range is
likely related to the variance in heating and/or cooling efficiency from halo
to halo. We demonstrate that these models are consistent with the observed
temperature and luminosity functions of clusters, if scatter in the
mass-luminosity relation is accounted for. Finally, we present predictions for
the redshift evolution of these scaling relations and luminosity/temperature
functions. Comparison with recent data at z<0.7 shows reasonable agreement with
a model that assumes a median entropy floor of K0=200 keV sq.cm. When
observations are extended to group scales (kT < 1 keV), this evolution will
have the potential to discriminate between an entropy floor that is independent
of redshift (for example, in a preheating scenario) and one that depends on the
cooling time of the halo.
