The inertial It\^o drift and its applications to particle collision

TL;DR

This paper investigates the limit behavior of inertial systems driven by Ornstein-Uhlenbeck forces, revealing an inertial-Itô drift that depends on the mass-to-correlation time ratio and applying it to particle collision phenomena.

Contribution

It introduces the inertial-Itô drift as a new limit term in inertial stochastic systems and explores its implications for particle dynamics and collisions in turbulent fluids.

Findings

Identification of inertial centrifugal and turbophoretic effects as inertial-Itô drifts.

Analysis of concentration effects linked to particle collision theory.

Clarification of the limit behavior depending on the mass-to-correlation time ratio.

Abstract

The small mass $\mu$ limit of an inertial system driven by an Ornstein Uhlenbeck fluid force, with correlation time $\epsilon$ going to zero, leads to a first order system with an additional drift, which we call inertial-It\^{o}-drift, depending on the limit $\alpha$ of the ratio $\mu/\epsilon$; the drift being zero when $\alpha=0$, corresponding to the Stratonovich integral in the limit equation, as in the Wong-Zakai theory, when applied directly to the first-order system with Ornstein-Uhlenbeck driver. We discuss the application of this result to particles driven by Stokes force;\ we identify inertial centrifugal effects and the so-called turbophoretic effect, as examples of the inertial-It\^{o}-drift. We also analyze concentration effects and their link with the theory of particle collision in turbulent fluids.