  Acceleration of cosmic-ray electrons (CRe) in the intra-cluster-medium (ICM)
is probed by radio observations that detect diffuse, Mpc-scale, synchrotron
sources in a fraction of galaxy clusters. Giant radio halos are the most
spectacular manifestations of non-thermal activity in the ICM and are currently
explained assuming that turbulence driven during massive cluster-cluster
mergers reaccelerates CRe at several GeV. This scenario implies a hierarchy of
complex mechanisms in the ICM that drain energy from large-scales into
electromagnetic fluctuations in the plasma and collisionless mechanisms of
particle acceleration at much smaller scales. In this paper we focus on the
physics of acceleration by compressible turbulence. The spectrum and damping
mechanisms of the electromagnetic fluctuations, and the mean-free-path (mfp) of
CRe are the most relevant ingredients that determine the efficiency of
acceleration. These ingredients in the ICM are however poorly known and we show
that calculations of turbulent acceleration are also sensitive to these
uncertainties. On the other hand this fact implies that the non-thermal
properties of galaxy clusters probe the complex microphysics and the weakly
collisional nature of the ICM.
