Scale-Dependent Velocity Fluctuations Generated by Molecular Collisions

A discrete binomial random-walk description of molecular collisions is used to quantify the variance of coarse-grained velocity fields arising solely from collision-induced momentum exchange. Closed-form expressions for the growth of velocity variance as functions of coarse-graining scale and time are derived and shown to imply a power-law decay of variance with averaging scale. Particle-based ensemble simulations validate the predicted scaling and temporal behaviour; surrogate ensemble tests demonstrate that phase/temporal coherence is required for the observed integrated transfer diagnostics. The analysis is intentionally restricted to collision-generated fluctuations in quiescent fluids and does not model cascade dynamics; implications for possible amplification under inertial dynamics are discussed cautiously. All data and the minimal verification instructions required to reproduce the summary tables are embedded in Appendix A.