  Compressive isothermal turbulence is known to have a near lognormal density
probability distribution function (PDF) with a width that scales with the sonic
Mach number and nature of the turbulent driving (solenoidal vs compressive).
However, the physical processes that mold the extreme high and low density
structures in a turbulent medium can be different, with the densest structures
being composed of strong shocks that evolve on shorter timescales than the low
density fluid. The density PDF in a turbulent medium exhibits deviations from
lognormal due to shocks, that increases with the sonic Mach number, which is
often ignored in analytic models for turbulence and star formation. We develop
a simple model for turbulence by treating it as a continuous Markov process,
which explains both the density PDF and the transient timescales of structures
as a function of density, using a framework developed in n Scannapieco &
Safarzadeh (2018). Our analytic model depends on only a single parameter, the
effective compressive sonic Mach number, and successfully describes the
non-lognormal behavior seen in both 1D and 3D simulations of supersonic and
subsonic compressive isothermal turbulence. The model quantifies the
non-lognormal distribution of density structures in turbulent environments, and
has application to star forming molecular clouds and star formation
efficiencies.
