  The time evolution of the probability density function (PDF) of the mass
density is formulated and solved for systems in free-fall using a simple
appoximate function for the collapse of a sphere. We demonstrate that a
pressure-free collapse results in a power-law tail on the high-density side of
the PDF. The slope quickly asymptotes to the functional form
$\mathrm{P}_v(\rho)\propto\rho^{-1.54}$ for the (volume-weighted) PDF and
$\mathrm{P}_m(\rho)\propto\rho^{-0.54}$ for the corresponding mass-weighted
distribution. From the simple approximation of the PDF we derive analytic
descriptions for mass accretion, finding that dynamically quiet systems with
narrow density PDFs lead to retarded star formation and low star formation
rates. Conversely, strong turbulent motions that broaden the PDF accelerate the
collapse causing a bursting mode of star formation. Finally, we compare our
theoretical work with observations. The measured star formation rates are
consistent with our model during the early phases of the collapse. Comparison
of observed column density PDFs with those derived from our model suggests that
observed star-forming cores are roughly in free-fall.
