  The turnover in the stellar initial mass function (IMF) at low mass suggests
the presence of two independent mass functions that combine in different ways
above and below a characteristic mass given by the thermal Jeans mass in the
cloud. In the random sampling model introduced earlier, the Salpeter IMF at
intermediate to high mass follows primarily from the hierarchical structure of
interstellar clouds, which is sampled by various star formation processes and
converted into stars at the local dynamical rate. This power law part is
independent of the details of star formation inside each clump and therefore
has a universal character. The flat part of the IMF at low mass is proposed
here to result from a second, unrelated, physical process that determines only
the probability distribution function for final star mass inside a clump of a
given mass, and is independent of both this clump mass and the overall cloud
structure. Both processes operate for all potentially unstable clumps in a
cloud, regardless of mass, but only the first shows up above the thermal Jeans
mass, and only the second shows up below this mass. Analytical and stochastic
models of the IMF that are based on the uniform application of these two
functions for all masses reproduce the observations well.
