Modeling Star Formation as a Markov Process in a Supersonic
  Gravoturbulent Medium
  Molecular clouds exhibit lognormal probability density functions (PDF) of
mass densities, which are thought to arise as a consequence of isothermal,
supersonic turbulence. Star formation is then widely assumed to occur in
perturbations in which gravitational collapse is faster than the rate of change
due to turbulent motions. Here we use direct numerical simulations to measure
this rate as a function of density for a range of turbulent Mach numbers, and
show it is faster at high densities than at low densities. Furthermore, we show
that both the density PDF and rate of change arise naturally in a simple model
of turbulence as a continuous Markov process. The one-dimensional Langevin
equation that describes this evolution depends on only two parameters, yet it
captures the full evolution seen in direct three-dimensional simulations. If it
is modified to include gravity, the Langevin equation also reproduces the rate
of material collapsing to high densities seen in turbulent simulations
including self-gravity. When generalized to include both temperature and
density, similar analyses are likely applicable throughout astrophysics.
