Second order Fermi reacceleration mechanisms and large scale synchrotron
  radio emission in intra-cluster bridges
  Radio observations at low frequencies with the Low Frequency Array (LOFAR)
start discovering gigantic radio bridges connecting pairs of massive galaxy
clusters. These observations probe unexplored mechanisms of in situ particle
acceleration that operate on volumes of several Mpc$^3$. Numerical simulations
suggest that such bridges are dynamically complex and that weak shocks and
super-Alfv\'{e}nic turbulence can be driven across the entire volume of these
regions. In this Letter we explore, for the first time, the role of second
order Fermi mechanisms for the reacceleration of relativistic electrons
interacting with turbulence in these peculiar regions. We assume the turbulent
energy flux measured in simulations and adopt a scenario in which relativistic
particles scatter with magnetic field lines diffusing in super-Alfv\'enic
turbulence and magnetic fields are amplified by the same turbulence. We show
that steep spectrum and volume filling synchrotron emission can be generated in
the entire intra-cluster bridge region thus providing a natural explanation for
radio bridges. Consequently, radio observations have the potential to probe the
dissipation of energy on scales larger than galaxy clusters and second order
Fermi mechanisms operating in physical regimes that are still poorly explored.
This has a potential impact on several branches of astrophysics and cosmology.
