On the stochastic nature of Galactic cosmic-ray sources

The precision measurements of the spectra of cosmic ray nuclei and leptons in recent years have revealed the existence of multiple features, such as the spectral break at $\sim 300$ GV rigidity seen by PAMELA and AMS-02 and more recently confirmed by DAMPE and CALET, the softening in the spectra of H and He nuclei at $\sim 10$ TV reported by DAMPE, confirming previous hints by NUCLEON and CREAM, a tiny change of slope at $\sim 40$ GeV in the electron spectrum, revealed by AMS-02, and the large spectral break at $\sim$ TeV reported by indirect (HESS, MAGIC and VERITAS) and direct (DAMPE, CALET) measurements of the total (electrons+positrons) lepton spectrum. In all these cases, the possibility has been suggested that these features might reflect the occasional presence of a local cosmic ray source, inducing a noticeable reshaping of the average expected spectra. All these proposals have to face the question of how likely it is for such a source to exist, a question that we address here in a quantitative way. We study the statistical properties of random distribution of sources in space and time, and the effect of the spiral structure of our Galaxy for both the spectra of light nuclei (p and He) and leptons (electrons and positrons) in different energy regions.