A Binomial Random-Walk Framework Quantifying Molecular Collision-Induced Fluctuations in Inertial-Range Eddy Formation

Continuum turbulence models rely on the premise that molecular motions average out at macroscopic scales (Knudsen number \(\:Kn\:\ll\:\:1\)). Viscosity damps sub-Kolmogorov molecular perturbations too quickly for inertial effects to act, leaving a gap in how finite disturbances arise at the Kolmogorov microscale. I close this gap by modeling each molecule as executing a three-dimensional binomial random walk. With step length \(\:l\) and collision rate \(\:n\), I derive \(\:⟨r⟩=\sqrt{\frac{8}{3\pi\:}}l\sqrt{n}\) which, for \(\:l\approx\:3\times\:10⁻¹⁰\) m and \(\:n\approx\:10¹²\) s⁻¹, yields ≈ 1 mm s ^- ¹ rms fluctuations. This matches colloidal diffusion in still water and shows that angstrom-scale molecular collisions produce a persistent rms velocity on the order of 1 mm s ^- ¹ at the Kolmogorov microscale. Continuum averaging (\(\:Kn\ll\:1\)) 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” \(\:Dₜₘ\:\approx\:\:\nu\:\:Re\). My results explain why the continuum assumption, although valid for mean transport, fails to capture the microscopic origins of turbulent fluctuations.