  The origin of hard X-ray (HXR) excess emission from clusters of galaxies is
still an enigma, whose nature is debated. One of the possible mechanism to
produce this emission is the bremsstrahlung model. However, previous analytical
and numerical calculations showed that in this case the intracluster plasma had
to be overheated very fast because suprathermal electrons emitting the HXR
excess lose their energy mainly by Coulomb losses, i.e., they heat the
background plasma. It was concluded also from these investigations that it is
problematic to produce emitting electrons from a background plasma by
stochastic (Fermi) acceleration because the energy supplied by external sources
in the form of Fermi acceleration is quickly absorbed by the background plasma.
In other words the Fermi acceleration is ineffective for particle acceleration.
We revisited this problem and found that at some parameter of acceleration the
rate of plasma heating is rather low and the acceleration tails of non-thermal
particles can be generated and exist for a long time while the plasma
temperature is almost constant. We showed also that for some regime of
acceleration the plasma cools down instead of being heated up, even though
external sources (in the form of external acceleration) supply energy to the
system. The reason is that the acceleration withdraws effectively high energy
particles from the thermal pool (analogue of Maxwell demon).
