Nonlinear theory of diffusive acceleration of particles by shock waves

Among the various acceleration mechanisms which have been suggested as resp onsible for the nonthermal particle spectra and associated radiation observ ed in many astrophysical and space physics environments, diffusive shock ac celeration appears to be the most successful. We review the current theoret ical understanding of this process, from the basic ideas of how a shock ene rgizes a few reactionless particles to the advanced nonlinear approaches tr eating the shock and accelerated particles as a symbiotic self-organizing s ystem. By means of direct solution of the nonlinear problem we set the limi t to the test-particle approximation and demonstrate the fundamental role o f nonlinearity in shocks of astrophysical size and lifetime. We study the b ifurcation of this system, proceeding from the hydrodynamic to kinetic desc ription under a realistic condition of Bohm diffusivity. We emphasize the i mportance of collective plasma phenomena for the global flow structure and acceleration efficiency by considering the injection process, an initial st age of acceleration and, the related aspects of the physics of collisionles s shocks. We calculate the injection rate for different shock parameters an d different species. This, together with differential acceleration resultin g from nonlinear large-scale modification, determines the chemical composit ion of accelerated particles. The review concentrates on theoretical and an alytical aspects but our strategic goal is to link the fundamental theoreti cal ideas with the rapidly growing wealth of observational data.