Diffusion processes of methane, water, and ethanol in poly(dimethylsiloxane) (PDMS) and polyethylene (PE) were investigated by molecular dynamics simulation. Pure liquid water and ethanol were also simulated. Simulations of 5 nanoseconds were performed for PDMS and PE with penetrant species, methane. The diffusion of methane in the polymer matrices exhibits anomalous (non-Einstein) behavior for time scales of 1 nanosecond. The excess chemical potentials of the penetrants in PDMS and the pure liquids were calculated by the Widom method. The excluded volume map sampling (EVMS) method and the continuum configurational bias (CCB) method were used to increase efficiency of sampling. Permeation rates, which were calculated from diffusion coefficients and solubilities, were in reasonable agreement with experiments.