Notes on the Stochastic Collection Equation
Résumé
The stochastic collection equation (SCE) describes the process that the cloud droplets grow in their sizes by their coalescence into larger sizes, and eventually to the size of rain droplets. This stochastic collection equation (SCE) is also referred to as Smoluchowski equation (1917) in honor of the Polish scientist, who derived this equation originally. The purpose of the present paper is to systematically examine the mathematical structures of the SCE system. Such an interest is also emerging from a different direction today (e.g., Xie 2024). Such a rather abstract approach is helpful for better interpreting the behaviors of the SCE simulated numerically , especially with more realistic parameters, especially in the form of the kernel. The present study will only consider some idealized forms of the kernel, the developed methodology of the analysis can, in principle, be applied to more complex and realistic forms of kernel. Such an abstract analysis also brings some basic insights of the growth process of the clouds, associated with the stochastic collection process. The paper consists of the four parts, in which the analysis proceeds from the simplest to more involved settings for elucidating different physical facets of the cloud evolution in stepwise manner.
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