We present a solution to Liouville's equation for an ensemble of charged particles propagating in magnetic fields. The solution is presented using an expansion in spherical harmonics of the phase space density, allowing a direct interpretation of the distribution of arrival directions of cosmic rays. The results are found for chosen conditions of variability and source distributions. We show there are two conditions for an initially isotropic flux of particles to remain isotropic while traveling through a magnetic field: isotropy and homogeneity of the sources. In case isotropically-distributed sources inject particles continuously in time, a transient magnetic induced dipole will appear. This dipole will vanish if the system reaches a steady-state. The formalism is used to analyze the data measured by the Pierre Auger Observatory, contributing to the understanding of the dependence of the dipole amplitude with energy and predicting the energy in which the quadrupole signal should be measured.

中文翻译：

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重新审视刘维尔定理对宇宙射线各向异性的影响

我们提出了在磁场中传播的带电粒子系综的刘维尔方程的解。该解决方案是使用相空间密度的球谐函数展开来提出的，从而可以直接解释宇宙射线到达方向的分布。结果是在选定的变异性和源分布条件下找到的。我们表明，最初各向同性的粒子通量在穿过磁场时保持各向同性需要两个条件：源的各向同性和均匀性。如果各向同性分布的源及时连续注入粒子，就会出现瞬态磁感应偶极子。如果系统达到稳态，该偶极子就会消失。该形式用于分析皮埃尔俄歇天文台测量的数据，有助于理解偶极子振幅与能量的依赖性，并预测应测量四极子信号的能量。