A continuous electron beam with a correlated emittance will eventually ther
malize. Initially, the beam emittance for an intense, high-brightness, spac
e-charge-dominated beam will oscillate, but after a sufficiently long time,
it will reach an equilibrium value. The emittance oscillations are due to
coherent transverse plasma oscillations in the beam and are a manifestation
of periodic energy exchange between potential and kinetic energies. The be
am eventually attains an equilibrium emittance, as the beam equipartitions
the kinetic and potential energies. This equipartioning is reached as the b
eam thermalizes due to a form of Landau damping of the radial oscillations
at different radial positions within the beam. Slight differences in the tr
ansverse plasma oscillation frequency for different radial positions lead t
o incoherence in the oscillations. In this paper, we calculate the equilibr
ium time scales required for equipartioning. We show that the equilibrium e
mittance scalings and magnitude can be predicted by conservation of energy
considerations. In addition, we show that, in the space-charge dominated re
gime,there is a correspondence between the energy-conservation approach and
the kinematic approach. [S1063-651X(99)10608-1].