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"Stochastic Particle Acceleration and the Problem of Background Plasma Overheating" D.O. Chernyshov and V.A. Dogiel (P.N.Lebedev Institute of Physics) Abstract: 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 background plasma. In other words the Fermi acceleration is ineffective for particle acceleration. We revisited this problem and found that at some parameters of acceleration the rate of plasma heating is rather low and the acceleration tails of non-thermal particles are generated and exist for a long time while the plasma temperature is almost constant . We showed also that for some regimes of acceleration the plasma is cooled down instead of to be heated up, though external sources (in the form of external acceleration) supply energy to the system. The reason is that in the last case the acceleration subtracts effectively high energy particles from the thermal pool that is the reason of plasma cooling (analogue of Maxwell's demon).