Based on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth's magnetosheath. We apply the statistical analysis using a Fokker-Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers- Moyal coefficient is linear and the second term is quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein-Uhlenbeck process. We demonstrate that the second order approximation of the Fokker-Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale-invariance, consistently with the stationary solution of the Fokker-Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.

翻译：基于磁层多尺度（MMS）任务，我们研究了地球磁鞘中的磁场涨落。应用Fokker-Planck方程进行统计分析，以探究导致空间等离子体随机涨落的过程。已知在流体力学尺度的惯性范围内，湍流呈现马尔可夫特征。我们将统计方法扩展到需应用动力学理论的更小空间尺度。本文详细研究并比较了弓激波后方、磁鞘内部及磁层顶附近的磁涨落特征。结果表明，在整个磁鞘中，第一Kramers-Moyal系数为磁增量的线性函数，第二项为二次函数，分别描述漂移和扩散。这应对应于Ornstein-Uhlenbeck过程的推广。我们证明Fokker-Planck方程的二阶近似导出概率密度函数的非高斯κ分布。在磁鞘的所有情况下，均恢复出近似幂律分布。对于某些中等尺度，κ分布呈现具有重尾的不同峰形。特别地，当κ参数较大时，该分布退化为标准高斯分布。值得注意的是，在较小的动力学尺度上，重标度分布表现出全局尺度不变性，与Fokker-Planck方程的稳态解一致。这些结果，尤其是动力学尺度上的发现，对于更深入理解空间及天体等离子体湍流系统的物理机制具有重要意义。