Introduction

The Probability Density Function of mass density is an important statistical property of the ISM that relates, for example, to gravitational collapse and star formation. Log-Normal PDFs have been discussed occasionally in both the cosmological and interstellar contexts (Hubble 1934, Peebles 1980, Ostriker 1984, Zinnecker 1984, Coles & Jones 1991). Vázquez-Semadeni (1994) noticed that the density PDFs in his 2-D numerical simulations of turbulence were consistent with a Log-Normal, and discussed possible reasons for the lognormality. Padoan et al. (1997) showed that the standard deviation of the Log-Normal PDFs in their isothermal 3-D simulations was approximately equal to half the rms Mach number. Scalo et al. (1998) raised the questions of how a polytropic equation of state, and more generally a realistic ISM cooling function, might influence the PDF.

In this contribution we investigate the question of the shape of the PDF for isothermal and polytropic equations of state, using analytical methods and by looking at results from 3-D simulations of supersonic turbulence.