  We propose a new, physically motivated fitting function for density PDFs in
turbulent gas. Although it is known that when gas is isothermal, the PDF is
approximately lognormal in the core, high-resolution simulations show large
deviations from exact log-normality. The proposed function provides an
extraordinarily accurate description of the density PDFs in simulations with
Mach numbers ~0.1-15 and dispersion in log(rho) from ~0.01-4 dex. Compared to a
lognormal or lognormal-skew-kurtosis model, the fits are improved by orders of
magnitude in the wings of the distribution (with fewer free parameters). This
is true in simulations using distinct numerical methods, including or excluding
magnetic fields. Deviations from lognormality are represented by a parameter T
that increases with the compressive Mach number. The proposed distribution can
be derived from intermittent cascade models of the longitudinal (compressive)
velocity differences, which should be directly related to density fluctuations,
and we provide a simple interpretation of the density PDF as the product of a
continuous-time relaxation process. As such the parameter T is consistent with
the same parameter needed to explain the (intermittent) velocity structure
functions; its behavior is consistent with turbulence becoming more
intermittent as it becomes more dominated by strong shocks. It provides a new
and unique probe of the role of intermittency in the density (not just
velocity) structure of turbulence. We show that this naturally explains some
apparent contradictory results in the literature based on use of different
moments of the density PDF, as well as differences based on whether
volume-weighted or mass-weighted quantities are measured. We show how these are
fundamentally related to the fact that mass conservation requires violations of
log-normal statistics.
