Coalescence of hydrometeors is commonly modeled using the Smoluchowski equation.  It is a mean-field equation that does not capture the stochastic nature of coalescence.  More exact methods include the DNS and the master equation.  The super-droplet method (SDM), which is a Lagrangian method for modeling cloud microphysics,  is another alternative. It is shown that the SDM with multiplicities equal to 1 is in agreement with the master equation.

Next, we use SDM simulations to determine validity of more approximate methods: the  Smoluchowski equation and the SDM with mulitplicities greater than 1. In the latter, we determine how many computational  droplets are necessary to correctly model the expected number and the standard deviation of the autoconversion time.

Then, SDM is used to study stochastic effects relevant for rain formation: fluctuations in the autoconversion time and lucky droplets.  Size of the coalescence cell is found to strongly affect system behavior. In small cells, correlations in droplet sizes and droplet depletion slow down rain formation.  In large cells, collisions between rain drops are more frequent and this also can slow down rain formation. The increase in the rate of collision between rain drops  may be an artefact caused by assuming a too large well-mixed volume. The highest ratio of rain water to cloud water is found in cells of intermediate sizes. Maximal size of a volume that is turbulently well-mixed with respect to coalescence is estimated at V = 1.5·10⁻² cm³.  The Smoluchowski equation is not valid in such small volumes. Implications for LES modeling are discussed.