Criterion for noise-induced synchronization: application to colloidal alignment

\def\S{synchroniz} \def\H{\Delta H} \def\D{dynamical system} \def\M{<\log|d\psi/d\phi|>} Asymmetric, self-assembled colloidal clusters can rotate stably as they descend under gravity. One may \S{}e a dispersion of copies of such a cluster using a force that randomly switches between two different directions[1]. This is an instance of ``noise-induced \S{}ation," demonstrated broadly in \D{}s that have a stable, periodic motion[2]. When such a system is perturbed by a prescribed transient force, it acquires a phase angle $\psi$ that depends on its initial phase $\phi$. For our colloidal dispersion the probability distribution of phases $\psi$ long after a switch in forcing is in general not uniform; thus the entropy $H$ of the ensemble has decreased. The phase map $\psi(\phi)$ provides strong constraints on the change $\H$ resulting from a switch: we show that the quantity $\M$ is an upper bound on $<\H>$. Thus whenever $\M~<0$, $H$ must decrease indefinitely on average. Our simulations show that this average is a good guide to the actual \S{}ation behavior. This bound and other properties of $\H$ apply broadly to any \D{} with a well-defined $\psi(\phi)$. [1] B. Moths, T. Witten. Phys Rev Lett, {\bf 110} 028301 (2013). [2] H. Nakao et al. Phys Rev E, {\bf 72} 026220 (2005).