On the probability density function of galactic gas.: I.: Numerical simulations and the significance of the polytropic index

We investigate the form of the one-point probability density 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 turbulence, of supersonic self-gravitating hydrodynamic turbulence, and of decaying Burgers turbulence produce in all cases filamentary density structures and evidence for a power-law density pdf at large densities with logarithmic slope between -1.7 and -2.3. This suggests that a power-law shape 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 one-dimensional 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 equal 1 if the Mach number is sufficiently small), while power laws develop for densities larger than the mean 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 should probably occur at densities around 10(3) and is possible at larger densities, depending strongly on the role of gas-grain heating and cooling. Several applications are examined. First, we question a recent derivation of the initial mass function from the density pdf by Padoan, Nordlund, & Jones because (1) the pdf does not contain spatial information and (2) 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.