Stochastic kinetic theory for collective behavior of hydrodynamically interacting active particles

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 two-dimensional systems, though we also show how superdiffusion of tracers in three dimensions can occur by our framework.

Figure 1

Fourier mode amplitudes of orientation correlation function, ${\text{log}}_{10}\left(N_p{\widehat{C}}_{o,\mathit{k}}\right)$, for pushers (a) and pullers (b) as a function of wave number $k$ and nondimensional rotational diffusivity ${\tilde{D}}_r$. The blank region in panel (a) corresponds to the region of linear instability of the pusher system. For both cases, the microswimmers are rodlike and slender ($\gamma =1$) with translational diffusivity of ${\tilde{D}}_t=0.45$.

Figure 2

Comparison of $N_p{\widehat{C}}_{o,\mathit{k}}-1$ between the numerical results from Fig. 1 and the slow-swimming perturbation approximation. The translational diffusivity is ${\tilde{D}}_t=0.45$, and the rotational diffusivity is ${\tilde{D}}_r=0.09$.

Figure 3

Fourier mode amplitudes of active stress correlation function, ${\text{log}}_{10}\left(N_p{\widehat{C}}_{s,\mathit{k}}{|}_{{\theta }_k=0}\right)$, for pushers (a) and pullers (b) as a function of wave number $k$ and nondimensional rotational diffusivity ${\tilde{D}}_r$. The blank region in panel (a) corresponds to the region of linear instability of the pusher system. For both cases, the microswimmers are rodlike and slender ($\gamma =1$) with translational diffusivity of ${\tilde{D}}_t=0.45$.

Figure 4

Comparison of $N_p{\widehat{C}}_{s,\mathit{k}}{|}_{{\theta }_k=0}$ between the numerical results from Fig. 3 and the slow-swimming perturbation approximation. The translational diffusivity is ${\tilde{D}}_t=0.45$, and the rotational diffusivity is ${\tilde{D}}_r=0.09$. The red line uses the first approximation (first term) from Eq. (19), while the blue line uses both terms shown in Eq. (19).

Figure 5

The diffusivity ${\tilde{D}}_{\mathrm{tr}}$ of non-Brownian passive point-particle tracers as a function of the rotational diffusivity of the microswimmer ${\tilde{D}}_r$. The symbols are from PDE-based stochastic simulations for slender rodlike pushers (squares) and pullers (circles). The lines are the solution to deterministic matrix equations that assume zero Kubo number. The solid red line is the calculation result for pushers, while the solid blue line is for pullers, and the solid black line is for noninteracting microswimmers.