Stochastic growth of cloud droplets by collisions during settling

In the last stage of droplet growth in clouds which leads to drizzle formation, larger droplets begin to settle under gravity and collide and coalesce with smaller droplets in their path. In this article, we shall deal with the simplified problem of a large drop settling amidst a population of identical smaller droplets. We present an expression for the probability that a given large drop suffers a given number of collisions, for a general statistically homogeneous distribution of droplets. We hope that our approach will serve as a valuable tool in dealing with droplet distribution in real clouds, which has been found to deviate from the idealized Poisson distribution due to mechanisms such as inertial clustering.