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We present a solution to Liouville's equation for an ensemble of chargedparticles propagating in magnetic fields. The solution is presented using anexpansion in spherical harmonics of the phase space density, allowing a directinterpretation of the distribution of arrival directions of cosmic rays. Theresults are found for chosen conditions of variability and sourcedistributions. We show there are two conditions for an initially isotropic fluxof particles to remain isotropic while traveling through a magnetic field:isotropy and homogeneity of the sources. In case isotropically-distributedsources inject particles continuously in time, a transient magnetic induceddipole will appear. This dipole will vanish if the system reaches asteady-state. The formalism is used to analyze the data measured by the PierreAuger Observatory, contributing to the understanding of the dependence of thedipole amplitude with energy and predicting the energy in which the quadrupolesignal should be measured.

Revisiting the implications of Liouville's theorem to the anisotropy of cosmic rays