Probability distribution functions (PDFs) and the corresponding cumulative distribution functions (CDFs) used in this study. Panel a shows the mass-weighted PDFs in sample simulation cells as a function of the sub-grid density n_H, where P_M(n_H) dn_H denotes the fraction of the total cell mass present at sub-grid densities in the range [n_H, n_H + dn_H]. The log-normal (Eq. (1)) and the log-normal+power-law (Eq. (3)) PDFs are shown in blue and red, respectively. The sample cells have a mean hydrogen density 〈n_H〉 = 100 cm^-3 (shown by the dotted black line). For the log-normal+power-law PDF, the transition density n_tr and the cut-off density n_cut are shown by the dashed and solid black lines, respectively. Panel b shows the corresponding CDFs, that is, C_M (n_H) denotes the fraction of the total cell mass present at sub-grid densities below n_H . For the log-normal+power-law PDF shown here, the power-law tail encloses ~91% of the total cell mass.