Monte Carlo and weighted-bin representations of the activation-energy distribution. (a) The particle Monte Carlo treatment assigns activation energies to implanted H atoms by stochastic sampling, which can sparsely represent low-probability portions of the distribution in low-flux surface elements. (b) The weighted activation-energy bin representation discretizes the same distribution into fixed probability-weighted bins, ensuring that all energy intervals contribute at each time step. (c), (d) Example surface abundance maps from the same model snapshot show that the weighted-bin field preserves the large-scale structure of the particle Monte Carlo field while reducing sampling noise. The maps use the same abundance scale and illumination geometry; the yellow circle marks the subsolar point, and white dashed lines mark the terminators.