A path-integral approach to coupled discrete-continuous problems

Typical cloud physics systems at small scales are often formulated as coupled discrete–continuous problems, comprising discrete, stochastically evolving hydrometeors and continuous, field-like thermodynamic variables. For modeling purposes, the inherent stochastic and particle-based nature of these systems is frequently simplified into more tractable mathematical frameworks, such as moment-based schemes. However, such approximations often fail to adequately capture the full impact of stochastic effects and the structure of distribution tails – features that can significantly influence system behavior. Although these effects can be resolved at small scales through numerical simulations of Master equations and related methods, approaches to upscale such descriptions to large-scale systems have remained elusive.
In this work, we introduce a novel mathematical framework that translates general coupled discrete–continuous problems into a path integral formulation, and consequently into an approximate field theory. This approach circumvents the need for computationally expensive numerical simulations and enables direct analytical computation of distribution moments. As a result, parameter spaces of models can be efficiently explored via analytical means, facilitating their application to significantly larger spatial and temporal scales.
We illustrate the efficacy of our method using a simple model system and explore its applicability to typical atmospheric situations.