  For over a century, raindrop size distributions have been a subject of
extensive scientific study, typically described by models including the
Marshall-Palmer exponential equation, gamma, Weibull, lognormal, and other
mathematical functions. In this work, we present a theory that integrates
deterministic principles from thermodynamics and fluid dynamics with stochastic
elements to predict expected raindrop diameters and describe ground-level
drop-size distributions. Importantly, our approach avoids assuming specific
drop-size dispersion processes or relying on multi-variable empirical data
fitting. We derive analytical equations for key raindrop parameters (e.g.,
median diameter, expected minimum and maximum diameters) and drop-size
distributions as functions of rainfall intensity. Our theoretical predictions
align well with extensive published experimental data, covering rainfall
intensities from 0.4 to 40 mm/h across diverse global locations. Additionally,
consistent with observations of super-large raindrops, our theory suggests a
maximum ground-level raindrop diameter limit of around 10 mm. We establish
analytical expressions for meteorological parameters like the Marshall-Palmer
constant and total raindrop concentration, validated against empirical data.
The theoretical approach presented here can find broad applications in climate
modeling, sprays and aerosol dynamics, bubbles and particles, extraterrestrial
rainfall, and paleoclimatology.
