Criterion for noise-induced synchronization: application to colloidal alignment

Abstract

Colloidal bodies of irregular shape rotate as they descend under gravity in solution. This rotational response provides a means of bringing a dispersion of identical bodies into a synchronized rotation with the same orientation using programmed forcing. We use the notion of statistical entropy to derive bounds on the rate of synchronization. These bounds apply generally to dynamical systems with stable periodic motion with a phase $ϕ(t)$, when subjected to an impulsive perturbation. The impulse causes a change of phase expressible as a phase map $ψ(ϕ)$. We derive an upper limit on the average change of entropy $\left<ΔH\right>$ in terms of this phase map; when this limit is negative, alignment must occur. For systems that have achieved a low entropy, the $\left<ΔH\right>$ approaches this upper limit.