  The thermodynamics of the diffuse, X-ray emitting gas in clusters of galaxies
is linked to the entropy level of the intra cluster medium. In particular,
models that successfully reproduce the properties of local X-ray clusters and
groups require the presence of a minimum value for the entropy in the center of
X-ray halos. Such a minimum entropy is most likely generated by
non-gravitational processes, in order to produce the observed break in
self-similarity of the scaling relations of X-ray halos. At present there is no
consensus on the level, the source or the time evolution of this excess
entropy. In this paper we describe a strategy to investigate the physics of the
heating processes acting in groups and clusters. We show that the best way to
extract information from the local data is the observation of the entropy
profile at large radii in nearby X-ray halos (z~0.1), both at the upper and
lower extremes of the cluster mass scale. The spatially and spectrally resolved
observation of such X-ray halos provides information on the mechanism of the
heating. We demonstrate how measurements of the size of constant entropy
(adiabatic) cores in clusters and groups can directly constrain heating models,
and the minimum entropy value. We also consider two specific experiments: the
detection of the shock fronts expected at the virial boundary of rich clusters,
and the detection of the isentropic, low surface-brightness emission extending
to radii larger than the virial ones in low mass clusters and groups. Such
observations will be a crucial probe of both the physics of clusters and the
relationship of non-gravitational processes to the thermodynamics of the
intergalactic medium.
