Mesoscale pattern formation of self-propelled rods with velocity reversal

We study self-propelled particles with velocity reversal interacting by uniaxial (nematic) alignment within a coarse-grained hydrodynamic theory. Combining analytical and numerical continuation techniques, we show that the physics of this active system is essentially controlled by the reversal frequency. In particular, we find that elongated, high-density, ordered patterns, called bands, emerge via subcritical bifurcations from spatially homogeneous states. Our analysis reveals further that the interaction of bands is weakly attractive and, consequently, bands fuse upon collision in analogy with nonequilibrium nucleation processes. Moreover, we demonstrate that a renormalized positive line tension can be assigned to stable bands below a critical reversal rate, beyond which they are transversally unstable. In addition, we discuss the kinetic roughening of bands as well as their nonlinear dynamics close to the threshold of transversal instability. Altogether, the reduction of the multiparticle system onto the dynamics of bands provides a unified framework to understand the emergence and stability of nonequilibrium patterns in this self-propelled particle system. In this regard, our results constitute a proof of principle in favor of the hypothesis in microbiology that velocity reversal of gliding rod-shaped bacteria regulates the transitions between various self-organized patterns observed during the bacterial life cycle.

Figure 1

Nematic order parameter [36] and corresponding snapshots for simulations of SPR, described by Eq. (1), for increasing reversal frequency $\lambda$. The crucial dependence of the degree of orientational order and stability of band structures on reversal is evident. Within high density bands, particles are nematically aligned. Parameters: $L_{x,y}=400,{\rho }_0=0.4,N=64000,D_{\phi }=0.07,\mu =\pi ,v_0=1,\beta \left(\mathbf{r}\right)=\mathrm{\Theta }\left(1-\left|\mathbf{r}\right|\right)/\pi ,\mathrm{\Delta }t=0.01$.

Figure 2

(a) Illustration of the potential $\mathcal{U}\left(B\right)$ for several values of $\gamma$ which determines stationary band solutions. (b) Band profiles $B\left(x\right)$ of periodicity $l_B=33$ (red solid line), $l_B=50$ (green dashed line), and $l_B=100$ (blue dotted line) that coexist at the density ${\overline{\rho }}_0=11/18$ [cf. Eq. (7)].

Figure 3

Bifurcations of band solutions: stable solutions are shown by solid lines, and linearly unstable solutions by dashed lines. Black lines represent the bifurcations of spatially homogeneous states. The mass of stable bands $M={\overline{\rho }}_0-5/18$ as predicted by the Schlögl model is indicated by a gray dotted line.

Figure 4

Illustration of a transversally modulated band. The dashed line indicates the filament $\zeta \left(y,t\right)$.

Figure 5

Phase diagram as a function of ${\overline{\rho }}_0\propto {\rho }_0\mu /D_{\phi }$ and the effective system size ${\overline{L}}_{x,y}=L_{x,y}\sqrt{32D_{\phi }(2\lambda +9D_{\phi })}/v_0$ for $\mathcal{D}=10$. Thick lines correspond to bifurcation points: the mean-field order-disorder transition for ${\overline{\rho }}_0=1/2$ is shown in green (dotted); the blue line (solid) indicates the transversal instability of the homogeneously ordered state [derivation not shown; cf. [35, 58, 59] and Eq. (5)]; the red (dashed) line shows the saddle-node bifurcation of bands as shown in Fig. 3 obtained by numerical continuation of the analytical band solution [Eq. (7)]. The parameter region corresponding to transversally unstable bands is shown in gray (TI) as predicted from the linearized filament dynamics [Eq. (10)]. Since bands are always transversally unstable for ${\overline{L}}_{x,y}\to \infty$ [17], the black solid line will span over the whole range of band existence (within the red, dashed lines).