  Some observations such as those presented in Walker et al. show that the
observed entropy profiles of the intra-cluster medium (ICM) deviate from the
power-law prediction of adiabatic simulations. This implies that
non-gravitational processes, which are absent in the simulations, may be
important in the evolution of the ICM, and by quantifying the deviation, we may
be able to estimate the feedback energy in the ICM and use it as a probe of the
non-gravitational processes. To address this issue we calculate the ICM entropy
profiles in a sample of 47 galaxy clusters and groups, which have been observed
out to at least $\sim r_{500}$ with Chandra, XMM-Newton and/or Suzaku, by
constructing a physical model to incorporate the effects of both gravity and
non-gravitational processes to fit the observed gas temperature and surface
brightness profiles via Bayesian statistics. After carefully evaluating the
effects of systematic errors, we find that the gas entropy profiles derived
with best-fit results of our model are consistent with the simulation-predicted
power-law profile near the virial radius, while the flattened profiles reported
previously can be explained by introducing the gas clumping effect, the
existence of which is confirmed in 19 luminous targets in our sample. We
calculate the total feedback energy per particle and find that it decreases
from $\sim 10$ keV at the center to about zero at $\sim 0.35$$r_{200}$ and is
consistent with zero outside $\sim 0.35$$r_{200}$, implying the upper limit of
the feedback efficiency $\sim 0.02$ for the super-massive black holes hosted in
the brightest cluster galaxies.
