An efficient Fokker-Planck solver and its application to stochastic particle acceleration in galaxy clusters

Particle acceleration by turbulence plays a role in many astrophysical environments. The nonlinear evolution of the underlying cosmic ray spectrum is complex and can be described by a Fokker-Planck equation, which in general has to be solved numerically. We present here an implementation to compute the evolution of a cosmic ray spectrum coupled to turbulence considering isotropic particle pitch-angle distributions and taking into account the relevant particle energy gains and losses. Our code can be used in runtime and post-processing to very large astrophysical fluid simulations. We also propose a novel method to compress cosmic ray spectra by a factor of 10, to ease the memory demand in very large simulations. We show a number of code tests, which firmly establish the correctness of the code. In this paper, we focus on relativistic electrons, but our code and methods can be easily extended to the case of hadrons. We apply our pipeline to the relevant problem of particle acceleration in galaxy clusters. We define a sub-grid model for compressible magnetohydrodynamic (MHD) turbulence in the intracluster medium and calculate the corresponding reacceleration time-scale from first principles. We then use a MHD simulation of an isolated cluster merger to follow the evolution of relativistic electron spectra and radio emission generated from the system over several Gyr.