Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, s, is well measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track s and [Formula: see text] in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of s and [Formula: see text] decay exponentially, as predicted by the model, and the decay timescale is ≈1/6 the eddy turnover time. The behaviors of the conditional averages of [Formula: see text] and [Formula: see text] are also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it.