  The underlying physics of giant radio halos and mini halos in galaxy clusters
is still an open question, which becomes more pressing with the growing number
of detections. In this paper, we explore the possibility that radio-emitting
electrons are generated in hadronic cosmic ray (CR) proton interactions with
ambient thermal protons of the intra-cluster medium. Our CR model derives from
cosmological hydrodynamical simulations of cluster formation and additionally
accounts for CR transport in the form of CR streaming and diffusion. This opens
the possibility of changing the radio halo luminosity by more than an order of
magnitude on a dynamical time scale. We build a mock galaxy cluster catalog
from the large MultiDark N-body LCDM simulation by adopting a phenomenological
gas density model for each cluster based on X-ray measurements that matches
Sunyaev-Zel'dovich (SZ) and X-ray scaling relations and luminosity function.
Using magnetic field strength estimates from Faraday rotation measure studies,
our model successfully reproduces the observed surface brightness profiles of
giant radio halos (Coma, A2163) as well as radio mini-halos (Perseus,
Ophiuchus), while obeying upper limits on the gamma-ray emission in these
clusters. Our model is also able to simultaneously reproduce the observed
bimodality of radio-loud and radio-quiet clusters at the same L_X as well as
the unimodal distribution of radio-halo luminosity versus the SZ flux Y;
thereby suggesting a physical solution to this apparent contradiction. For a
plausible fraction of 10% radio-loud clusters, our model matches the NVSS
radio-halo luminosity function. Constructing an analytical radio-halo
luminosity function, we demonstrate the unique prospects for low-frequency
radio surveys (such as the LOFAR Tier 1 survey) to detect ~3500 radio halos
back to redshift two and to probe the underlying physics of radio halos.
[abridged]
