  With the recently emerging global interest in building a next generation of
circular electron-positron colliders to study the properties of the Higgs
boson, and other important topics in particle physics at ultra-high beam
energies, it is also important to pursue the possibility of implementing
polarized beams at this energy scale. It is therefore necessary to set up
simulation tools to evaluate the beam polarization at these ultra-high beam
energies. In this paper, a Monte-Carlo simulation of the equilibrium beam
polarization based on the Polymorphic Tracking Code(PTC) (Schmidt et al., 2002
[1]) is described. The simulations are for a model storage ring with parameters
similar to those of proposed circular colliders in this energy range, and they
are compared with the suggestion (Derbenev et al., 1978 [2]) that there are
different regimes for the spin dynamics underlying the polarization of a beam
in the presence of synchrotron radiation at ultra-high beam energies. In
particular, it has been suggested that the so-called "correlated" crossing of
spin resonances during synchrotron oscillations at current energies, evolves
into "uncorrelated" crossing of spin resonances at ultra-high energies.
