  [Abridged] Since 1971 observations in X rays of thousands galaxy clusters
have uncovered huge amounts of hot baryons filling up the deep gravitational
potential wells provided by dark matter (DM) halos with sizes of millions
light-years and masses of some 10^15 M_sun. At temperatures T~10^8 K and with
average densities of n~1 particle per liter, such baryons add up to some 10^14
M_sun. With the neutralizing electrons, they constitute the best
proton-electron plasma in the Universe (Intra Cluster Plasma, ICP). A key
physical feature of the ICP is constituted by its good local Thermal
equilibrium, and by its overall hydrostatic condition in the DM wells,
modulated by entropy. The latter is set up in the cluster center by the initial
halo collapse, and is progressively added at the outgrowing cluster boundary by
standing shocks in the supersonic flow of intergalactic gas into the DM wells.
We review these entropy-based models and discuss their outcomes and predictions
concerning the ICP observables in X rays and in microwaves. The results provide
a baseline for disentangling a number of additional and intriguing physical
processes superposed to the general equilibrium. We cover: the central entropy
erosion produced by radiative cooling vs. the intermittent energy inputs mainly
due to active galactic nuclei and mergers; outer turbulent support linked with
weakening shocks and decreasing inflow through the virial boundary, causing
reduced entropy production; the development from high to low entropy levels
throughout a typical cluster; perturbations of the equilibrium up to outright
disruption due to deep impacts of infalling galaxy groups or collisions with
comparable companion clusters; relativistic energy distributions of electrons
accelerated during such events, producing extended radio emission by
synchrotron radiation, and contributing to non-thermal pressure support for the
ICP.
