Scale-Dependent Velocity Fluctuations Generated by Molecular Collisions

TL;DR

This paper develops a mathematical model to describe how velocity fluctuations caused by molecular collisions depend on scale and time, validated by simulations, with implications for understanding fluid dynamics.

Contribution

It introduces a discrete binomial random-walk model for collision-induced velocity fluctuations and derives closed-form expressions for their scale and time dependence.

Findings

Velocity variance decays as a power law with scale.

Simulation results confirm the predicted scaling behavior.

Coherence is necessary for observed transfer diagnostics.

Abstract

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.