  Hadron yields in high energy heavy ion collisions have been fitted and
reproduced by thermal models using standard statistical distributions. These
models give insight into the freeze-out conditions at varying beam energies. In
this paper we investigate changes to this analysis when the statistical
distributions are replaced by Tsallis distributions for hadrons. We investigate
the appearance of near-thermal equilibrium state at SPS and RHIC energies. We
obtain better fits with smaller chi^2 for the same hadron data, as applied
earlier in the thermal fits for SPS energies but not for RHIC energies. This
result indicates that at RHIC energies the final state is very well described
by a single freeze-out temperature with very little room for fluctuations.
