Detailed Balance and Exact Results for Density Fluctuations in
  Supersonic Turbulence
  The probabilistic approach to turbulence is applied to investigate density
fluctuations in supersonic turbulence. We derive kinetic equations for the
probability distribution function (PDF) of the logarithm of the density field,
$s$, in compressible turbulence in two forms: a first-order partial
differential equation involving the average divergence conditioned on the flow
density, $\langle \nabla \cdot {\bs u} | s\rangle$, and a Fokker-Planck
equation with the drift and diffusion coefficients equal to $-\langle {\bs u}
\cdot \nabla s | s\rangle$ and $\langle {\bs u} \cdot \nabla s | s\rangle$,
respectively. Assuming statistical homogeneity only, the detailed balance at
steady state leads to two exact results, $\langle \nabla \cdot {\bs u} | s
\rangle =0$, and $\langle {\bs u} \cdot \nabla s | s\rangle=0$. The former
indicates a balance of the flow divergence over all expanding and contracting
regions at each given density. The exact results provide an objective criterion
to judge the accuracy of numerical codes with respect to the density statistics
in supersonic turbulence. We also present a method to estimate the effective
numerical diffusion as a function of the flow density and discuss its effects
on the shape of the density PDF.
