  We explore the relationship between the thermal Sunyaev-Zel'dovich (tSZ)
power spectrum amplitude and the halo mass and redshift of galaxy clusters in
South Pole Telescope (SPT) data, in comparison with three $N$-body simulations
combined with semi-analytical gas models of the intra-cluster medium.
Specifically, we calculate both the raw and fractional power contribution to
the full tSZ power spectrum amplitude at $\ell = 3000$ from clusters as a
function of halo mass and redshift. We use nine mass bins in the range $1
\times 10^{14}\ M_\odot\ h^{-1} < M_{500} < 2 \times 10^{15}\ M_\odot\ h^{-1}$,
and two redshift bins defined by $0.25 < z < 0.59$ and $0.59 < z < 1.5$. We
additionally divide the raw power contribution in each mass bin by the number
of clusters in that bin, as a metric for comparison of different gas models. At
lower masses, the SPT data prefers a model that includes a mass-dependent bound
gas fraction component and relatively high levels of AGN feedback, whereas at
higher masses there is a preference for a model with a lower amount of feedback
and a complete lack of non-thermal pressure support. The former provides the
best fit to the data overall, in regards to all metrics for comparison. Still,
discrepancies exist and the data notably exhibits a steep mass-dependence which
all of the simulations fail to reproduce. This suggests the need for additional
mass- and redshift-dependent adjustments to the gas models of each simulation,
or the potential presence of contamination in the data at halo masses below the
detection threshold of SPT-SZ. Furthermore, the data does not demonstrate
significant redshift evolution in the per-cluster tSZ power spectrum
contribution, in contrast to self-similar model predictions.
