For the last 50 years since the introduction of the basic transport equation by Parker (1965), models for the acceleration and propagation of energetic particles in the heliosphere, the Galaxy and beyond, have adhered to the paradigm of a diffusion-advection equation describing the evolution of the particle distribution function. For many years the linear Parker equation has been employed with great success, quantitatively computing the differential intensities of energetic test particles without altering the properties of the background plasma. In the 1990s, first extensions to self-consistency were made in two directions, with energetic particles changing the dynamics of the thermal plasma they traverse and affecting the evolution of the turbulence spectra within. The desired self-consistency is achieved by a nonlinear coupling of the linear Parker equation (possibly extended to contain momentum diffusion) to the (magneto)hydrodynamical equations describing a thermal background plasma or to a wave or turbulence transport equation. However, in such coupling schemes, the nonlinearity does not directly extend to the diffusion process itself, which remains treated as normal (Gaussian) diffusion within the linear Parker equation.
More recent analyses of observations of energetic electrons and protons as well as of the magnetic fluctuations upstream of interplanetary shocks and of relativistic electrons in supernova remnants imply that the transport of cosmic-ray particles in the presence of turbulent scattering may not be consistent with Gaussian diffusion but rather be superdiffusive. An alternative to the above-mentioned solution of coupled systems involving a linear transport equation for energetic particles is the consideration of a single, but nonlinear diffusion-advection equation. Interestingly, it has also been shown that the transport resulting from such nonlinear treatment can be anomalous. It appears that the topics of nonlinearity and anomaly of energetic particle transport as well as acceleration are intrinsically linked.
Both these properties represent conceptual, mathematical, and numerical challenges, which will be addressed during the workshop.
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