Phenomenological explanation for the shape of the spectrum of cosmic rays with energies <Emphasis Type="Italic">E</Emphasis> > 10 GeV

Assuming that the energy gain by cosmic-ray (CR) particles is a stochastic process with stationary increments, we derive expressions for the shape of their energy spectrum up to energies E ～ 10¹⁸ eV. In the ultrarelativistic case under study, the energy is proportional to the momentum, whose time derivative is the force. According to the Fermi mechanism, a particle accelerates when it passes through a system of shock waves produced by supernova explosions. Since these random forces act on time scales much shorter than the particle lifetime, we assume them to be delta-correlated in time. In this case, due to the linear energy-momentum relationship, the mean square of the energy (increments) is proportional to the differential scale τ(E) ～ Eτ(≥E), where τ (≥E) is the cumulative time it takes for a particle to gain an energy ≥E. The probability of finding a particle with energy ≥E somewhere in the system is inversely proportional to the time it takes to gain the energy E. To estimate an upper limit for the space number density of CR particles, we use estimates of the CR volume energy density, a quantity known for our Galaxy. It is taken to be constant in the range 10 GeV ≤ E ≤ 3 × 10⁶ GeV, where the index of the energy spectrum was found to be ?8/3 ≈ ?2.67 against its empirical value of ?2.7. In the range 3 × 10⁶ GeV ≤ E < 10⁹ GeV, the upper limit for the volume energy density is estimated by using the results from the previous range to be ?28/9 ≈ ?3.11 against its empirical value of ?3.1. The numerical coefficients in the suggested shapes of the spectrum can be determined by comparison with observational data. Thus, the CR energy spectrumis the result of the random walks of ultrarelativistic particles in energy/momentum space caused by the Fermi mechanism.