  We compare equilibrium probability distributions obtained from Monte Carlo
simulations for different spin exchange dynamics with the exact Boltzmann
distribution for the fixed magnetization Ising model on small lattices. We
present simple arguments and numerical evidence in order to show that
"efficient" Kawasaki exchange does not lead to Boltzmann equilibrium
distributions and that nearest-neighbour equal spin-exchange must be
considered. Arbitrary distance opposite spin interchange is indicated as an
alternative to obtaining the full phase diagram with phase separation.
