Maximum Energy of Particles in Plasmas

Particles are accelerated to very high, non-thermal energies in space, solar, and astrophysical plasma environments. In cosmic ray physics, the "Hillas limit" is often used as a rough estimate (or the necessary condition) of the maximum energy of particles. This limit is based on the concepts of one-shot direct acceleration by a system-wide motional electric field, as well as stochastic and diffusive acceleration in strongly turbulent environments. However, it remains unclear how well this limit explains the actual observed maximum energies of particles. Here we show, based on a systematic review, that the observed maximum energy of particles -- those in space, solar, astrophysical, and laboratory environments -- often reach the energy predicted by the Hillas limit. We also found several exceptions, such as electrons in solar flares and jet-terminal lobes of radio galaxies, as well as protons in planetary radiation belts, where deviations from this limit occur. We discuss possible causes of such deviations, and we argue in particular that there is a good chance of detecting ultra-high-energy ($\sim$100 GeV) solar flare electrons that have not yet been detected. We anticipate that this study will facilitate further interdisciplinary discussions on the maximum energy of particles and the underlying mechanisms of particle acceleration in diverse plasma environments.