  In galaxy clusters the thermal Sunyaev-Zel'dovich (SZ) effect from the hot
intracluster medium (ICM) provides a direct, self-contained measure of the
pressure integrated over crossing lines of sight, that is intrinsically
independent of redshift and well suited for evolutionary studies. We show in
detail how the size of the effect and its pattern on the sky plane are directly
related to the entropy levels in the ICM, and how they characterize the cluster
cores and outskirts independently. We find that at redshifts z<0.3 the signals
to be expected in the cores considerably exceed those detected at 10'
resolution with the Planck satellite. We propose that at 1' resolutions as
implemented on recent ground instrumentation for mapping features in individual
clusters, the average patterns of the SZ signals can provide a direct and
effective way to find and count cool, low-entropy cores and hot, high-entropy
outskirts out to z~2. Such counts will tell the timing and the mode of the
processes that drive the evolution of the ICM from the distant to the local
cluster population.
