Numerical simulations are performed of dispersion and polydispersity of particles in isotropic incompressible turbulence. The mass loading of the particles is assumed to be small; thus the effects of particles on turbulence is neglected (one-way coupling). A stochastic model is employed to simulate the carrier phase. The results of the simulations are compared with direct numerical simulation (DNS) data and theoretical results. The stochastic model predicts most of the trends as portrayed by DNS and theory. However, the continuity effect associated with the crossing trajectories effect is not captured. Also, the peaking in the variation of the particle asymptotic diffusivity coefficient with the particle time constant is not observed. For evaporating particles, the stochastic model predicts thinner probability density functions (pdfs) for the particle diameter as compared with DNS generated pdfs. The model is implemented to investigate the effects of gravity on evaporation. It is shown that the depletion rate increases with increase of the drift velocity at short and intermediate times, but an opposite trend is observed at long times. The standard deviation and skewness of the particle diameter indicate peak values in their variations with the drift velocity. Dispersion of evaporating particles decreases with respect to that of non-evaporating particles at small drift velocities; an opposite trend is observed at large drift velocities. The effects of the initial evaporation rate and the particle Schmidt number on the evaporation in the gravity environment are also studied.