  The most important evidences for non-thermal phenomena in galaxy clusters
comes from the spectacular synchrotron radio emission diffused over Mpc scale
observed in a growing number of massive clusters. A promising possibility to
explain giant radio halos is given by the presence of relativistic electrons
reaccelerated by some kind of turbulence generated in the cluster volume during
merger events. With the aim to investigate the connection between thermal and
non-thermal properties of the ICM, in this paper we develope a statistical
magneto-turbulent model which describes in a self-consistent way the evolution
of the thermal ICM and that of the non-thermal emission from clusters. Making
use of the extended Press & Schechter formalism, we follow cluster mergers and
estimate the injection rate of the fluid turbulence generated during these
energetic events. We then calculate the evolution of the spectrum of the
relativistic electrons in the ICM during the cluster life by taking into
account both the electron-acceleration due to the merger-driven turbulence and
the relevant energy losses of the electrons. We end up with a synthetic
population of galaxy clusters for which the evolution of the ICM and of the
non-thermal spectrum emitted by the accelerated electrons is calculated. The
generation of detectable non-thermal radio and hard X-ray emission in the
simulated clusters is found to be possible during major merger events for
reliable values of the model parameters. In addition the occurrence of radio
halos as a function of the mass of the parent clusters is calculated and
compared with observations. In this case it is found that the model
expectations are in good agreement with observations.
