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 { \boldsymbol { u } } |s \rangle , and a Fokker-Planck equation with the drift and diffusion coefficients equal to - \langle { \boldsymbol { u } } \cdot \nabla s|s \rangle and \langle { \boldsymbol { 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 { \boldsymbol { u } } |s \rangle = 0 , and \langle { \boldsymbol { 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 .