Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order

We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations, not parallel to the ordering direction, at zero wave number. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature 503, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability.

Figure 1

A log-log plot of the dimensionless decay rates for the two eigenmodes associated with the coupled dynamics of the total density and angular fluctuations in a weakly compressible system [Eq. (9)]. While both modes display a diffusive (slope 2) relaxation for small dimensionless wave vector $\tilde{q}$, taking the dimensionless compressibility $\tilde{\chi }$ to zero shifts the blue curve to the left, implying that the relaxation rate associated with any finite $\tilde{q}$ goes to infinity. Meanwhile, the second relaxation rate (black curve) develops a wide $\tilde{q}$-independent plateau, mimicking the nonhydrodynamic relaxation rate associated with a truly incompressible system.