Anisotropic equipartition state in a charged particle beam

The purpose of this Letter is to conjecture a characterization of the anisotropic equipartition state. The anisotropic equipartition state is defined through a phase space density which is uniform on the invariant surface of ξ, where ξ is the ratio between the oscillation energies in the x- and y-directions. It is a version of the ergodic hypothesis where the invariant surface of ξ plays the role of the conserved energy. We show that the anisotropic equipartition state is characterized by the following properties: the development of an elliptical shape with increasing size along the x-direction, the presence of a coupling between transversal emittances, halo formation along a preferential direction, stationarity of the temperature and a growth of the entropy in the cascade form.