# Swimmer suspensions on substrates: anomalous stability and long-range order

## TL;DR

This paper develops a comprehensive theory for two-dimensional suspensions of polar active particles on substrates, revealing conditions for anomalous stability or instability, and predicting long-range order and flocking transitions without banding.

## Contribution

It introduces a unified theoretical framework that explains stability, fluctuations, and phase transitions in active polar particle suspensions on substrates, correcting previous misconceptions.

## Key findings

- Polar order can be anomalously stable or unstable depending on a key parameter.
- The theory predicts long-range order in 2D polar suspensions, contrary to some earlier claims.
- A flocking transition occurs without the banding instability previously thought necessary.

## Abstract

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 at zero wavenumber. 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 ${\bf 503}$, 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