“Super GZK” Particles in a Classic Kramers’ Diffusion-over-a-barrier Model. I. The Case of Protons

In interactions of ultra-high-energy cosmic ray (UHECR) protons with cosmic microwave background photons, we focus in this work on photopion production reactions and the effects of the measured, broad, energy-loss distributions in these reactions on the evolution of the protons’ density functions in energy space. We rely on a Fokker–Planck transport equation in energy space whose transport coefficients are calculated using laboratory measurements. We also derive a Fokker–Planck potential that accounts for both systematic (drift) and stochastic (dispersive) energy losses due to photopion production reactions. Our results show that dispersive energy losses have significant effects on estimating the protons’ horizon distance and their energy spectrum, as well as to elucidate a broadness in the GZK cutoff. We use the derived Fokker–Planck potential to assign a characteristic probability for a proton to clear the potential barrier as a function of energy. This estimate of probability can be used to assist observations in distinguishing between protons and heavy ions as charged particles. Our model is able to account for the so-called super GZK particles as a classic diffusion-over-a-barrier manifestation of the transport of UHECR protons in energy space in quantifying the extent and broadness of the GZK cutoff.