  We investigate the form of the one-point probability distribution function
(pdf) for the density field of the interstellar medium using numerical
simulations that successively reduce the number of physical processes included.
Two-dimensional simulations of self-gravitating supersonic MHD and hydrodynamic
turbulence, and of decaying Burgers turbulence, produce in all cases
filamentary density structures and a power-law density pdf with logarithmic
slope around -1.7. This suggests that the functional form of the pdf and the
general filamentary morphology are the signature of the nonlinear advection
operator. These results do not support previous claims that the pdf is
lognormal. A series of 1D simulations of forced supersonic polytropic
turbulence is used to resolve the discrepancy. They suggest that the pdf is
lognormal only for effective polytropic indices $\gamma=1$ (or nearly lognormal
for $\gamma\not=1$ if the Mach number is sufficiently small), while power laws
develop at high densities if $\gamma<1$. We evaluate the polytropic index for
conditions relevant to the cool interstellar medium using published cooling
functions and different heating sources, finding that a lognormal pdf may occur
at densities between 10$^3$ and at least 10$^4$ cm$^{-3}$. Several applications
are examined. First, we question a recent derivation of the IMF from the
density pdf by Padoan, Nordlund & Jones because a) the pdf does not contain
spatial information, and b) their derivation produces the most massive stars in
the voids of the density distribution. Second, we illustrate how a distribution
of ambient densities can alter the predicted form of the size distribution of
expanding shells. Finally, a brief comparison is made with the density pdfs
found in cosmological simulations.
