  The statistical mechanics of phase transitions in dense systems of
polydisperse particles presents distinctive challenges to computer simulation
and analytical theory alike. The core difficulty, namely dealing correctly with
particle size fractionation between coexisting phases, is set out in the
context of a critique of previous simulation work on such systems. Specialized
Monte Carlo simulation techniques and moment free energy method calculations,
capable of treating fractionation exactly, are then described and deployed to
study the fluid-solid transition of an assembly of repulsive spherical
particles described by a top-hat "parent" distribution of particle sizes. The
cloud curve delineating the solid-fluid coexistence region is mapped as a
function of the degree of polydispersity $\delta$, and the properties of the
incipient "shadow" phases are presented. The coexistence region is found to
shift to higher densities as $\delta$ increases, but does not exhibit the sharp
narrowing predicted by many theories and some simulations.
