  A key parameter to the description of all star formation processes is the
density structure of the gas. In this letter, we make use of probability
distribution functions (PDFs) of Herschel column density maps of Orion B,
Aquila, and Polaris, obtained with the Herschel Gould Belt survey (HGBS). We
aim to understand which physical processes influence the PDF shape, and with
which signatures. The PDFs of Orion B (Aquila) show a lognormal distribution
for low column densities until Av 3 (6), and a power-law tail for high column
densities, consistent with a rho r^-2 profile for the equivalent spherical
density distribution. The PDF of Orion B is broadened by external compression
due to the nearby OB stellar aggregates. The PDF of a quiescent subregion of
the non-star-forming Polaris cloud is nearly lognormal, indicating that
supersonic turbulence governs the density distribution. But we also observe a
deviation from the lognormal shape at Av>1 for a subregion in Polaris that
includes a prominent filament. We conclude that (i) the point where the PDF
deviates from the lognormal form does not trace a universal Av-threshold for
star formation, (ii) statistical density fluctuations, intermittency and
magnetic fields can cause excess from the lognormal PDF at an early cloud
formation stage, (iii) core formation and/or global collapse of filaments and a
non-isothermal gas distribution lead to a power-law tail, and (iv) external
compression broadens the column density PDF, consistent with numerical
simulations.
