Microscopic Derivation of Stochastic Thin-Film Equation and Extensions to Two-Layer and Evaporating Films

Abstract

While classical thin-film hydrodynamics models widely govern macroscopic fluid layers, they fail to capture the molecular-level thermal fluctuations that dictate nanoscale interfacial dynamics. Standard stochastic extensions rely on the long-wave approximation of the continuum fluctuating Navier-Stokes equations; however, this top-down paradigm lacks a formal connection to microscopic particle trajectories and cannot readily accommodate complex, non-equilibrium boundary mechanisms. We resolve this limitation by deriving the stochastic thin-film equation directly from the exact underlying N-particle Hamiltonian dynamics. This is achieved via a systematic, bottom-up coarse-graining framework based on a generalized fluctuating dynamic density-functional theory. Crucially, this first principles formulation offers unparalleled flexibility, enabling the seamless integration of multi-layered interfaces and non-conserved dynamics such as evaporation. For these complex configurations, our framework rigorously yields entirely novel stochastic thin-film equations featuring emergent noise terms that encapsulate microscopic fluctuations missing from classical top-down models.