  Radio galaxies play an important role in the seeding of cosmic rays and
magnetic fields in galaxy clusters. Here, we simulate the evolution of
relativistic electrons injected into the intracluster medium by radio galaxies.
Using passive tracer particles added to magnetohydrodynamical adaptive-mesh
simulations, we calculate the evolution of the spectrum of relativistic
electrons taking into account energy losses and re-acceleration mechanisms
associated with the dymamics of the intracluster medium. Re-acceleration can
occur at shocks via diffusive shock acceleration, and in turbulent flows via
second-order Fermi re-acceleration. Relativistic electrons from radio galaxies
are found to fill the intracluster medium over scales of several $100 \rm
~Myr$, and they create a stable reservoir of fossil electrons which remains
available for further re-acceleration by shock waves and turbulent gas motions.
In the near future, deep radio observations (especially at low frequencies) are
likely to probe such mechanisms in galaxy clusters.
