Theory and simulations of particle acceleration in collisionless shock waves (or pulses) are described.
Particle simulations show that in a multi-ion-species plasma with hydrogen the major component, a shock wave can reflect some of the hydrogen ions and accelerate them to high energies [1,2] (and references therein). At the same time, it accelerates all the heavy ions [3]. The maximum speeds of the heavy ions are nearly the same; i.e., independent of particle species. Also, electrons can be accelerated to ultrarelativistic energies such that the Lorentz factor γ>100 [4]. These explain basic properties of solar energetic particles; prompt acceleration of ions to 1-10 GeV and electrons to 10-100 MeV, with the elemental composition of high-energy heavy ions similar to that of the solar corona.
Furthermore, if nonthermal energetic particles are present in a plasma, they can be accelerated to much higher energies [5]; ion acceleration from γ=4 to γ=160 was observed in test particle simulations, in which particle orbits were calculated by use of electromagnetic fields that were obtained from self-consistent particle simulations. Particle simulations also show that a shock wave in an electron-positron-ion plasma can accelerate positrons to very high energies [6]; γ=600 was reported.
By studying the structure of nonlinear magnetosonic waves as well as particle motions, a coherent quantitative theory for these five different acceleration mechanisms has been developed.

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