Emergent length-scale in microswimmer suspensions*

Recent years witnessed a significant interest in physical, biological and engineering properties of self-propelled particles, such as bacteria or synthetic microswimmers. One of the most striking features of interacting microswimmers is the appearance of collective motion: at densities high enough, the system is characterised by jets and vortices comprising many individual swimmers. Although many experimental and theoretical works have shown the appearance of a length-scale intrinsic to the ensuing collective flow, its precise origin is not understood.

In this work, we investigate the statistical properties of self-propelling particles with hydrodynamic interactions. Starting from the kinetic theory of microswimmers, we derive a closed set of mean-field moment equations. Performing large-scale pseudo-spectral simulations, we calculate the corresponding energy spectra, and spatial correlations for various values of the mean particle density. Our results demonstrate the emergence of a typical length-scale in the collective phase and we show that it is set by the microswimmer run-length and the inter-particle distance. This length-scale determines the size of strongly correlated regions: it is infinite at the point of the mean-field transition to collective motion, and decreases with increasing microswimmer density. At large scales, the system effectively behaves as a gas of non-interacting swimmers.