Information-optimal mixing at low Reynolds number

TL;DR

This paper introduces a universal measure of mixing efficiency at low Reynolds number using mutual information, derives optimal shear protocols, and reveals their time-reversal symmetry and minimal energy cost for information erasure.

Contribution

It provides an exact solution for optimal mixing protocols based on mutual information, applicable to general shear flows with energy constraints.

Findings

Optimal protocols are universal and time-reversal symmetric.

Derived a compact expression for mutual information in shear flows.

Established a minimum energetic cost for erasing information in non-equilibrium systems.

Abstract

Mutual information between particle positions before and after mixing provides a universal assumption-free measure of mixing efficiency at low Reynolds number which accounts for the kinematic reversibility of the Stokes equation. For a generic planar shear flow with time-dependent shear rate, we derive a compact expression for the mutual information as a nonlinear functional of the shearing protocol and solve the associated extremisation problem exactly to determine the optimal control under both linear and non-linear constraints, specifically total shear and total dissipation per unit volume. Remarkably, optimal protocols turn out to be universal and time-reversal symmetric in both cases. Our results establish a minimum energetic cost of erasing information in a broad class of non-equilibrium drift-diffusive systems.