The probability distribution function of column density in molecular clouds

We discuss the probability distribution function (PDF) of column density re sulting from density fields with lognormal PDFs, applicable to isothermal g as (e.g., probably molecular clouds). For magnetic and nonmagnetic numerica l simulations of compressible, isothermal turbulence forced at intermediate scales (1/4 of the box size), we find that the autocorrelation function (A CF) of the density field decays over relatively short distances compared to the simulation size. We suggest that a "decorrelation length" can be defin ed as the distance over which the density ACF has decayed to, for example, 10% of its zero-lag value, so that the density "events" along a line of sig ht can be assumed to be independent over distances larger than this, and th e central limit theorem should be applicable. However, using random realiza tions of lognormal fields, we show that the convergence to a Gaussian is ex tremely slow in the high-density tail. As a consequence, the column density PDF is not expected to exhibit a unique functional shape, but to transit i nstead from a lognormal to a Gaussian form as the ratio eta of the column l ength to the decorrelation length (i.e., the number of independent events i n the cloud) increases. Simultaneously, the variance of the PDF decreases. For intermediate values of eta, the column density PDF assumes a nearly exp onential decay. For cases with a density contrast of 10(4), as found in int ermediate-resolution simulations, and expected from giant molecular clouds (GMCs) to dense molecular cores, the required value of eta for convergence to a Gaussian is at least a few hundred, or, for 10(6), several thousand. W e then discuss the density power spectrum and the expected value of eta in actual molecular clouds, concluding that they are uncertain since they may depend on several physical parameters. Observationally, our results suggest that eta may be inferred from the shape and width of the column density PD F in optically thin line or extinction studies. Our results should also hol d for gas with finite-extent power-law underlying density PDFs, which shoul d be characteristic of the diffuse, nonisothermal neutral medium (with temp eratures ranging from a few hundred to a few thousand degrees). Finally, we note that for eta greater than or similar to 100, the dynamic range in col umn density is small (less than a factor of 10), but this is only an averag ing effect, with no implication on the dynamic range of the underlying dens ity distribution.