Statistical consistency of a discrete collisional model for velocity fluctuations and stochastic transport

Abstract

A discrete collisional model for velocity fluctuations is developed and statistically validated to establish consistency between molecular-scale randomness and continuum stochastic transport. The formulation is based on a compound Poisson representation of collision-induced velocity increments, yielding an analytically tractable linear variance growth law. Under diffusive normalization, the model converges toward Gaussian statistics, consistent with central limit behavior. Numerical simulations confirm linear variance growth, decay of skewness toward zero, and relaxation of kurtosis toward the Gaussian value. Consistency with continuum fluctuating hydrodynamics is further demonstrated through a single-mode Ornstein–Uhlenbeck response, which exhibits inverse-viscosity scaling of stationary variance in agreement with theoretical prediction. The results provide explicit validation of the statistical pathway from discrete molecular collisions to Gaussian stochastic forcing and linear hydrodynamic response within a minimal and computationally transparent framework.