  This paper examines the transition from isotropic to anisotropic beam
profiles in a linear focusing channel. Considering a high-intensity ion beam in
space-charge dominated regime and large mismatched {\it RMS} beam size
initially, observe a fast anisotropy situation of the beam, characterized for a
transition of the transversal section round to elliptical with a coupling of
transversal emittance driven for instabilities of nonlinear space-charge
forces. Space-charge interactions in high-intensity linear accelerator can lead
to equipartitioning of energy between the degrees of freedom. The
anisotropization phenomena suggest a kind of route to equipartition. In order
to understand the initial dynamical behavior of an anisotropic beam, in
particular, to study possible mechanisms of equipartition connected with phase
space we have to know how we can compute the variables (volume, area of
surface, and area projected) that characterize the anisotropic beam in phase
space. The purpose of this paper is to propose one definiton of the anisotropic
equipartition \cite{Yankov1}. Anisotropic equipartition corresponds to a phase
space density uniform on the surface invariant of the $\xi$, a version of the
ergodic hypothesis where the $\xi$ invariant play the role of the conserved
energy \cite{Kandrup3}. In the state of anisotropic equipartition, the beam
temperature is stationary, the entropy grows in the cascade form, there is a
coupling of transversal emittance, the beam develops an elliptical shape with a
increase in its size along one direction and there is halo formation along one
direction preferential.
