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

Pure molecular velocity increments are individually short-lived and are viscously dissipated on timescales many orders of magnitude shorter than Kolmogorov turnover times, so the classical continuum assumption that molecular-scale fluctuations are dynamically irrelevant is untenable when asking how perturbations are seeded at dissipation scales. Treating molecular collisions as a three-dimensional binomial random walk shows that zero-mean collision steps accumulate Brownian-style variance that survives continuum averaging (which removes the mean but preserves second moments), producing a finite rms velocity at dissipation scales on the order of mm/s. These irreducible microscale fluctuations at the Kolmogorov scale provide a persistent population of seed perturbations. In high–Reynolds-number flows, Lagrangian strain and positive finite-time Lyapunov exponents selectively and exponentially amplify those seeds that experience favorable stretching, converting them into microscale eddies that then participate in the upscale cascade to fully developed inertial-range eddies and enhanced eddy diffusivity. The analysis, therefore, connects angstrom-scale collisional physics to macroscopic turbulence and indicates that molecular-seed statistics should be incorporated into closure models and validated with targeted DNS/MD and micro-PIV experiments.