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