Diffusion in momentum space as a picture of second-order Fermi   acceleration
  Energetic particles in a turbulent medium can be subject to second-order Fermi acceleration due to scattering on moving plasma waves. This mechanism leads to growing particle momentum dispersion and, at the same time, increases the mean particle energy. In the most frequently met situations both processes can be represented by a single momentum diffusion term in the particle kinetic equation. In the present paper we discuss the conditions allowing the additional term for regular acceleration to arise. For forward-backward asymmetric scattering centres, besides the diffusive term one should explicitly consider the regular acceleration term in momentum space, which can consist of the first-order ($\propto V$), as well as the second-order ($\propto V^2$) part in the wave velocity V . We derive the condition for the scattering probability in the wave rest frame required for vanishing the regular acceleration term and provide a simple mechanical example illustrating the theoretical concepts. Finally, we address its possible role in cosmic ray acceleration processes. 
M. Ostrowski (1,2) and G. Siemieniec-Ozi\c{e}b{\l}o(1) (1 -   Astronomical Observatory of the Jagiellonian University, Craco...