We discuss the probability distribution function ( PDF ) of column density resulting from density fields with lognormal PDFs , applicable to isothermal gas ( e.g. , probably molecular clouds ) .
For magnetic and non-magnetic numerical simulations of compressible , isothermal turbulence forced at intermediate scales ( 1/4 of the box size ) , we find that the autocorrelation function ( ACF ) of the density field decays over relatively short distances compared to the simulation size .
We suggest that a “ decorrelation length ” can be defined 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 sight can be assumed to be independent over distances larger than this , and the Central Limit Theorem should be applicable .
However , using random realizations of lognormal fields , we show that the convergence to a Gaussian is extremely 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 instead from a lognormal to a Gaussian form as the ratio \eta of the column length to the decorrelation length ( i.e. , the number of independent events in the cloud ) increases .
Simultaneously , the PDF ’ s variance decreases .
For intermediate values of \eta , the column density PDF assumes a nearly exponential decay .
For cases with a density contrast of 10 ^ { 4 } ( resp . 10 ^ { 6 } ) , as found in intermediate-resolution simulations , and expected from GMCs to dense molecular cores , the required value of \eta for convergence to a Gaussian is at least a few hundred ( resp .
several thousand ) .
We 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 PDF in optically-thin-line or extinction studies .
Our results should also hold for gas with finite-extent power-law underlying density PDFs , which should be characteristic of the diffuse , non-isothermal neutral medium ( temperatures ranging from a few hundred to a few thousand degrees ) .
Finally , we note that for \eta \gtrsim 100 , the dynamic range in column density is small ( \lesssim a factor of 10 ) , but this is only an averaging effect , with no implication on the dynamic range of the underlying density distribution .