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We investigate the form of the one-point probability distribution function(pdf) for the density field of the interstellar medium using numericalsimulations that successively reduce the number of physical processes included.Two-dimensional simulations of self-gravitating supersonic MHD and hydrodynamicturbulence, and of decaying Burgers turbulence, produce in all casesfilamentary density structures and a power-law density pdf with logarithmicslope around -1.7. This suggests that the functional form of the pdf and thegeneral filamentary morphology are the signature of the nonlinear advectionoperator. These results do not support previous claims that the pdf islognormal. A series of 1D simulations of forced supersonic polytropicturbulence is used to resolve the discrepancy. They suggest that the pdf islognormal only for effective polytropic indices $\gamma=1$ (or nearly lognormalfor $\gamma\not=1$ if the Mach number is sufficiently small), while power lawsdevelop at high densities if $\gamma<1$. We evaluate the polytropic index forconditions relevant to the cool interstellar medium using published coolingfunctions and different heating sources, finding that a lognormal pdf may occurat densities between 10$^3$ and at least 10$^4$ cm$^{-3}$. Several applicationsare examined. First, we question a recent derivation of the IMF from thedensity pdf by Padoan, Nordlund & Jones because a) the pdf does not containspatial information, and b) their derivation produces the most massive stars inthe voids of the density distribution. Second, we illustrate how a distributionof ambient densities can alter the predicted form of the size distribution ofexpanding shells. Finally, a brief comparison is made with the density pdfsfound in cosmological simulations.Comment: Submitted to ApJ. 23 pages, 15 postscript figure

On the Probability Density Function of Galactic Gas. I. Numerical Simulations and the Significance of the Polytropic Index