  Caustics in the dynamics of heavy particles in turbulence accelerate particle
collisions. The rate $\mathscr{J}$ at which these singularities form depends
sensitively on the Stokes number St, the non-dimensional inertia parameter.
Exact results for this sensitive dependence have been obtained using Gaussian
statistical models for turbulent aerosols. However, direct numerical
simulations of heavy particles in turbulence yield much larger
caustic-formation rates than predicted by the Gaussian theory. In order to
understand possible mechanisms explaining this difference, we analyse a
non-Gaussian statistical model for caustic formation in the limit of small St.
We show that at small St, $\mathscr{J}$ depends sensitively on the tails of the
distribution of Lagrangian fluid-velocity gradients. This explains why
different authors obtained different St-dependencies of $\mathscr{J}$ in
numerical-simulation studies. The most-likely gradient fluctuation that induces
caustics at small St, by contrast, is the same in the non-Gaussian and Gaussian
models. Direct-numerical simulation results for particles in turbulence show
that the optimal fluctuation is similar, but not identical, to that obtained by
the model calculations.
