Stochastic kinetic theory for collective behavior of hydrodynamically interacting active particles

Abstract

Self-propelled particles with hydrodynamic interactions (microswimmers) have previously been shown to produce long-range ordering phenomena. Many theoretical explanations for these collective phenomena are connected to instabilities in the hydrodynamic or kinetic equations. By incorporating stochastic fluxes into the mean field kinetic equation, we quantify the dynamics of a suspension of microswimmers in the parameter regime where the deterministic equation is stable. We can thereby compute nontrivial collective phenomena concerning spatial correlations of orientation and stress as well as the enhanced diffusion of tracer particles. Our analysis here focuses primarily on twodimensional systems, though we also show how superdiffusion of tracers in three dimensions can occur by our framework.

Swimming Suppresses Correlations in Dilute Suspensions of Pusher Microorganisms

“…We have overcome a significant technical difficulty in including particle selfpropulsion into a theory that goes beyond the mean-field assumption and explicitly accounts for correlations between microswimmers. This difficulty has limited previous theoretical work on this problem to either the case of shaker microswimmers [42] or the case of swimming being subdominant compared to the translational thermal diffusion [92]. The only theory to date that has accounted for arbitrary swimming speeds was developed by Nambiar, Garg, and Subramanian [91], who analytically considered pairwise correlations between microswimmers; i.e., their results are OðΔ 2 Þ accurate.…”

“…A systematic account for strong correlations between all microswimmers is achieved by Stenhammar et al [42], who develop a kinetic theory for suspensions of "shakers"-particles that apply forces to the fluid but do not self-propel. A similar theory is developed by Qian, Kramer, and Underhill [92], who study a stochastic kinetic theory for two-dimensional suspensions of swimming microorganisms. Analytical results obtained in that work are limited to the case of slow swimming-a perturbation theory that assumes that microswimmer self-propulsion is a small effect compared to their thermal diffusion and advection by other microswimmers.…”

Symmetric Mixtures of Pusher and Puller Microswimmers Behave as Noninteracting Suspensions

“…In spite of its simplicity, the dipolar description of microswimmers has been shown to quantitatively describe the enhanced diffusion of passive tracer particles in E. coli suspensions [8,11,12], as well as qualitatively explaining the onset of "active turbulence," whereby suspensions of bacteria undergo a transition to collective swimming characterized by significantly enhanced fluid velocities and long-ranged flow fields [7,[13][14][15][16][17][18][19]. Importantly, the transition to active turbulence as well as the buildup of pretransitional swimmer-swimmer correlations strongly depend on the sign of the force dipole [10,20], where active turbulence is only present for rearactuated "pusher" microswimmers such as most bacteria. Their front-actuated counterpart, "puller" microswimmers, are less common in nature: the bacterium Caulobacter crescentus is able to switch between pusher and puller propulsion modes [21], and the front-actuated alga Chlamydomonas oscillates between pusher and puller modes during its flagellar beat cycle [22,23].…”

“…A suspension of swimming microorganisms, such as bacteria or algae, is one of the archetypal examples of biological active matter at the microscopic scale [1][2][3][4][5]. At dilute concentrations, direct collisions between swimmers are rare, and interactions in biological microswimmer suspensions are therefore dominated by long-ranged hydrodynamic interactions leading to complex collective behavior and significant swimmer-swimmer correlations [3,[6][7][8][9][10]. Arguably, the simplest description of biological microswimmers is that of a force dipole acting on the fluid, leading to a flow field that decays as the inverse square of the distance from the organism.…”

“…While the instability leading to active turbulence can be inferred from a mean-field treatment, in order to capture the dynamics at intermediate microswimmer densities, but still below n c , it is necessary to go beyond the mean-field description to include the effect of swimmerswimmer correlations. Recent efforts [10,20,36] have shown that correlations between microswimmers become significant at concentrations far below n c . The mean-field description is thus only accurate for very dilute suspensions, where microswimmers can be described as effectively noninteracting, and pushers and pullers become statistically equivalent.…”