  Increasing evidences provided primarily by the cluster lensing and numerical
simulations of cluster formation indicate that galaxy clusters may contain
compact cores that are substantially smaller than the cores revealed by the
X-ray observations of hot intracluster gas. In this paper we present a model
that describes how two distinct cores can grow simultaneously as a result of
infall from the background dark matter. This model needs a pre-existing
localized large fluctuation, which can be the non-Gaussian density peak of the
primordial fluctuations, to seed the individual growing cluster. On the other
hand, numerous recent observations also show that no strong evolution is
detected for galaxy clusters within the redshift up to $z\sim 0.8$. We
therefore present a comparison of observations with the saturated cluster
configuration resulting from our infall model. In the saturated state, the
predicted compact core mass is about few$\times 10^{11} M_\odot$ and the core
size about 1 kpc for a cluster mass $\sim$ several $\times 10^{15} M_\odot$
within a radius of 3 Mpc. We have successfully reproduced the dark matter
distribution revealed by the gravitational lensing, and the observed radial
distributions of cluster galaxies, intracluster gas (i.e., the conventional
$\beta$ model) and baryon fractions in a consistent way. This model, when
combined with the observed X-ray surface brightness profiles of clusters,
predicts that the overall temperature of intracluster gas has a tendency of
radial decline with a mean polytropic index $\gamma\approx1.2$. Finally, the
so-called $\beta$ discrepancy finds a natural explanation for such a polytropic
gas in our model.
