  We demonstrate that the time-dependent projected Gross-Pitaevskii equation
derived earlier [Davis, et al., J. Phys. B 34, 4487 (2001)] can represent the
highly occupied modes of a homogeneous, partially-condensed Bose gas. We find
that this equation will evolve randomised initial wave functions to
equilibrium, and compare our numerical data to the predictions of a gapless,
second-order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33,
3847 (2000)]. We find that we can determine the temperature of the equilibrium
state when this theory is valid.
  Outside the range of perturbation theory we describe how to measure the
temperature of our simulations. We also determine the dependence of the
condensate fraction and specific heat on temperature for several interaction
strengths, and observe the appearance of vortex networks. As the
Gross-Pitaevskii equation is non-perturbative, we expect that it can describe
the correct thermal behaviour of a Bose gas as long as all relevant modes are
highly occupied.
