Traveling bands, clouds, and vortices of chiral active matter

We consider stochastic dynamics of self-propelled particles with nonlocal normalized alignment interactions subject to phase lag. The role of the lag is to indirectly generate chirality into particle motion. To understand large-scale behavior, we derive a continuum description of an active Brownian particle flow with macroscopic scaling in the form of a partial differential equation for a one-particle probability density function. Due to indirect chirality, we find a spatially homogeneous nonstationary analytic solution for this class of equations. Our development of kinetic and hydrodynamic theories towards such a solution reveals the existence of a wide variety of spatially nonhomogeneous patterns reminiscent of traveling bands, clouds, and vortical structures of linear active matter. Our model may thereby serve as the basis for understanding the nature of chiral active media and designing multiagent swarms with designated behavior.

Figure 1

Examples of homogeneous particle dynamics according to Eq. (1). (a) Linear polar (Vicsek) phase with $\alpha =0$. (b) Nonlocalized chimera state at consecutive time points $t_1<t_2<t_3$ with $\alpha =1.3$ (see the Supplemental Material [50, 51] for a corresponding movie). (c) Disordered motion with $\alpha =1.6$. Particles are colored according to their instantaneous orientations. White arrows indicate the mean direction of motion. Other parameters: $v_0=1.0,\sigma =1.0,\varrho =0.01,D_{\phi }=0.01,N=5\times {10}^3$.

Figure 2

Description of homogeneous particle dynamics. (a) Particle phase distributions for the dynamics presented in Fig. 1. The main figure shows the distributions for respective snapshots in Fig. 1. The insets show the respective distributions for particle dynamics in Figs. 1 and 1. (b) Evolution of the polar order parameter magnitude over time starting from random initial conditions for different values of phase lag $\alpha$. The curves show its mean value out of 100 experiments, with standard deviation shown as shaded regions. Model parameters are the same as in Fig. 1.

Figure 3

Solution of the system of self-consistent equations (see Appendix pp3) comprising a DF as a traveling wave solution (3) and a complex order parameter defined by that solution. (a) The order parameter magnitude $R$ and (b) the group velocity $v$ versus the phase lag $\alpha$ and the noise strength $D_{\phi }$. The dark gray line indicates the order-disorder transition line $D_{\phi }=\frac{1}{2}cos\alpha$. The critical group velocity along that line is $v=-\frac{1}{2}sin\alpha$.

Figure 4

Examples of a spatially homogeneous traveling wave solution (3). (a) DFs for the different values of the phase lag $\alpha$. (b) The circular skewness of Eq. (3) versus the phase lag and the noise, quantified by a circular skewness coefficient $\widehat{s}=\mathbb{E}\left[sin2(\phi -\mathrm{\Theta })\right]/{\left(1-\left|\mathbb{E}\left[e^{i\phi }\right]\right|\right)}^{3/2}$ [44]. Crosses indicate the parameter values used to generate the DFs in (a). The dark gray line indicates the order-disorder transition line, to compare with Fig. 3.

Figure 5

Phase diagrams in the parameter space of (a) the noise strength ${\widehat{D}}_{\phi }$ and the phase lag $\alpha$, and (b) the rescaled radius of interaction $\varrho /{\widehat{v}}_0$ and the phase lag, as predicted by the kinetic theory (see Appendix pp4). Color shows the maximal real part of the strongest unstable mode. The gray line is the order-disorder transition line ${\widehat{D}}_{\phi }=\frac{1}{2}cos\alpha$. Above the line in (a), spatially homogeneous disordered motion (SHDM) is stable; below the line in (a) and to the left in (b), spatially homogeneous ordered motion (SHOM) given by Eq. (3) is stable in the black region. Gray crosses indicate parameter values, selected to exemplify particle dynamics in Fig. 6.

Figure 6

Particle dynamics according to Eq. (1) (see the Supplemental Material [50, 51] for corresponding movies). Abbreviations stand for a dense cloud (DC), a traveling band (TB), a multiheaded chimera (MHC), a stationary vortex (SV), a dynamic vortex (DV), a momentum wave (MW), and a spatially nonhomogeneous momentum (SNM). Color indicates the direction of motion in the HSV color map. Each particle is represented as an elongated object over several time points. Parameter values correspond to those marked on the phase diagrams in Fig. 5. Parameters: $N=5\times {10}^4,L=1,\tilde{\varrho }=1,{\widehat{v}}_0=0.01$, (a) $\varrho =0.01,\alpha =0.78,{\widehat{D}}_{\phi }=0.2075$, (b) $\varrho =0.01,\alpha =0.9,{\widehat{D}}_{\phi }=0.18$, (c) $\varrho =0.01,\alpha =1.3,{\widehat{D}}_{\phi }=0.06$, (d) $\varrho =0.01,\alpha =1.45,{\widehat{D}}_{\phi }=0.01$, (e) $\varrho =0.4,\alpha =1.45,{\widehat{D}}_{\phi }=0.005$, (f) $\varrho =0.2,\alpha =1.36,{\widehat{D}}_{\phi }=0.005$, (g) $\varrho =0.01,\alpha =1.3,{\widehat{D}}_{\phi }=0.02$, (h) $\varrho =0.01,\alpha =1.0,{\widehat{D}}_{\phi }=0.0375$, (i) $\varrho =0.01,\alpha =1.0,{\widehat{D}}_{\phi }=0.0575$, and (j) $\varrho =0.01,\alpha =1.07,{\widehat{D}}_{\phi }=0.145$.

Figure 7

Coarse-grained marginal density function $\rho (r,t)$ (the upper row) and the direction of a momentum field $w(r,t)$ (the lower row). Parameters: $N=5\times {10}^4,\tilde{\varrho }=1,v_0=0.01$, (a) $\varrho =0.01,\alpha =0.78,D_{\phi }=0.2075$, (b) $\varrho =0.01,\alpha =0.9,D_{\phi }=0.18$, (c) $\varrho =0.01,\alpha =1.3,D_{\phi }=0.06$, (d) $\varrho =0.01,\alpha =1.45,D_{\phi }=0.01$, and (e) $\varrho =0.4,\alpha =1.45,D_{\phi }=0.005$.

Figure 8

Instability regions given by the analysis of the hydrodynamic equations (E53). The gray line denotes the order-disorder transition line $D_{\phi }=\frac{1}{2}cos\alpha$. The colored lines are found analytically by restricting the analysis to longitudinal perturbations solely. They enclose a line, below which long-wavelength perturbations acting on the traveling wave solution appear. The blue (yellow) line is defined by Eq. (E76) with $v=-sin\alpha$ ($v=-\frac{1}{2}sin\alpha$). The violet and yellow regions are obtained by numerically solving an eigenvalue problem for Eq. (E62). The violet region includes parameters for which at most one Fourier mode becomes unstable for each wave vector. The yellow region includes parameters for which there are at most two Fourier modes that become unstable for each wave vector. Other parameters are $\tilde{\varrho }=1,\varrho =0.01,v_0=0.01$.

Figure 9

Dispersion relations $\lambda (k_x,k_y)=\underset{n\in \mathbb{Z}}{max}{\lambda }_n(k_x,k_y)$ obtained by solving an eigenvalue problem for Eq. (D46). Grey markers indicate wave vectors, at which there is a Fourier mode with a positive real part. Parameters: $N=5\times {10}^4,\tilde{\varrho }=1,v_0=0.01$, (a) $\varrho =0.01,\alpha =0.78,D_{\phi }=0.2075$, (b) $\varrho =0.01,\alpha =0.9,D_{\phi }=0.18$, (c) $\varrho =0.01,\alpha =1.3,D_{\phi }=0.06$, (d) $\varrho =0.01,\alpha =1.45,D_{\phi }=0.01$, (e) $\varrho =0.4,\alpha =1.45,D_{\phi }=0.005$, (f) $\varrho =0.2,\alpha =1.36,D_{\phi }=0.005$, (g) $\varrho =0.01,\alpha =1.3,D_{\phi }=0.02$, (h) $\varrho =0.01,\alpha =1.0,D_{\phi }=0.0375$, (i) $\varrho =0.01,\alpha =1.0,D_{\phi }=0.0575$, and (j) $\varrho =0.01,\alpha =1.07,D_{\phi }=0.145$.

Figure 10

Positive real parts of each dispersion relation from Fig. 9. Parameters are the same as in Fig. 9.