  When very small particles are suspended in a fluid in motion, they tend to
follow the flow. How such tracer particles are mixed, transported, and
dispersed by turbulent flow has been successfully described by statistical
models. Heavy particles, with mass densities larger than that of the carrying
fluid, can detach from the flow. This results in preferential sampling,
small-scale fractal clustering, and large collision velocities. To describe
these effects of particle inertia, it is necessary to consider both particle
positions and velocities in phase space. In recent years, statistical
phase-space models have significantly contributed to our understanding of
inertial-particle dynamics in turbulence. These models help to identify the key
mechanisms and non-dimensional parameters governing the particle dynamics, and
have made qualitative, and in some cases quantitative predictions. This article
reviews statistical phase-space models for the dynamics of small, yet heavy,
spherical particles in turbulence. We evaluate their effectiveness by comparing
their predictions with results from numerical simulations and laboratory
experiments, and summarise their successes and failures. Annu. Rev. Fluid Mech.
56: In press. DOI: 10.1146/annurev-fluid-032822-014140.
