A Collision-Based Kinetic Model for the Molecular Seeding and Amplification of Turbulence

Abstract Continuum turbulence models rely on the premise that molecular motions average out at macroscopic scales (Knudsen number \(\:Kn\:\:\:1\)). I overturn this view by showing that angstrom-scale molecular collisions produce a persistent rms velocity on the order of 1 mm s⁻¹ at the Kolmogorov microscale. Modeling each molecule as executing a three-dimensional binomial random walk—with step length \(\:l\) and collision rate \(\:n\)—I derive \(\:⟨r⟩=√8/3π\:l√n\) which, for \(\:l≈\:3×\:10⁻¹⁰\) m and \(\:n≈\:10¹²\) s⁻¹, yields ≈ 1 mm s⁻¹ rms fluctuations—matching colloidal diffusion in still water. Continuum averaging (\(\:Kn\:1\)) can eliminate only the zero-mean component of each step but preserves its variance, so these molecular “noise” seeds endure. In high-Reynolds-number flows, inertial advection and positive Lyapunov exponents exponentially amplify this fixed noise floor into the full inertial-range cascade, giving rise to macroscopic eddies and enhanced “eddy diffusivity” \(\:Dₜₘ\:≈\:\:ν\:\:Re\). Our results explain why the continuum assumption, although valid for mean transport, fails to capture the microscopic origins of turbulent fluctuations.