  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 dynamics of the intracluster medium. Re-acceleration can
occur at shocks via diffusive shock acceleration, and in turbulent flows via
second-order Fermi re-acceleration. This study confirms that relativistic
electrons from radio galaxies can efficiently fill the intracluster medium over
scales of several $100 \rm ~Myr$, and that they create a stable reservoir of
fossil electrons that remains available for further re-acceleration by shock
waves and turbulent gas motions. Our results also show that late evolution of
radio lobes and remnant radio galaxies is significantly affected by the
dynamics of the surrounding intracluster medium. Here the diffusive
re-acceleration couples the evolution of relativistic particles to the gas
perturbations. In the near future, deep radio observations, especially at low
frequencies, can probe such mechanisms in galaxy clusters.
