Gibbs states and Brownian models for coexisting haze and cloud droplets

Cloud microphysics studies include how tiny cloud droplets grow, and become rain. This is crucial for understanding cloud properties like size, lifespan, and impact on climate through radiative effects. Small, weak-updraft clouds near the haze-to-cloud transition are especially difficult to measure and understand. They are abundant but hard to capture by satellites. K\"ohler's theory explains initial droplet growth but struggles with large particle groups. Here, we present a stochastic, analytical framework building on K\"ohler's theory to account for (monodisperse) aerosols and cloud droplets interaction through competitive growth in a limited water vapor field. These interactions are modeled by sink terms while fluctuations in supersaturation affecting droplet growth are modeled by nonlinear, white noise terms. Our results identify hysteresis mechanisms in the droplet activation and deactivation processes. Our approach allows for multimodal cloud's droplet size distributions supported by lab experiments, offering a new perspective on haze-to-cloud transition and small cloud formation.