Derivation of a generalized Kappa distribution from the scaling properties of solar wind magnetic field fluctuations at kinetic scales

Context. Kinetic-scale dynamics in weakly collisional space plasmas usually exhibits a self-similar statistics of magnetic field fluctuations. This implies the existence of an invariant probability density function (master curve). Aims. We provide an analytical derivation of the master curve by assuming that perpendicular fluctuations can be modeled through a scale-dependent Langevin equation. Methods. In our model, magnetic field fluctuations are the stochastic variable, and their scale-to-scale evolution is assumed to be a Langevin process. We propose a formal derivation of the master curve describing the statistics of the fluctuations at kinetic scales. The model predictions were tested on independent data samples of the fast solar wind measured near the Sun by Parker Solar Probe and near the Earth by Cluster. Results. The master curve is a generalization of the Kappa distribution with two parameters: One parameter regulates the tails, and the other controls the asymmetry. The model predictions match the spacecraft observations up to 5σ and even beyond in the case of perpendicular magnetic field fluctuations.