Fluctuations and pattern formation in self-propelled particles

We consider a coarse-grained description of a collection of self-propelled particles given by hydrodynamic equations for the density and polarization fields. We find that the ordered moving or flocking state of the system is unstable to spatial fluctuations beyond a threshold set by the self-propulsion velocity of the individual units. In this region, the system organizes itself into an inhomogeneous state of well-defined propagating stripes of flocking particles interspersed with low-density disordered regions. Further, we find that even in the regime where the homogeneous flocking state is stable, the system exhibits large fluctuations in both density and orientational order. We study the hydrodynamic equations analytically and numerically to characterize both regimes.

Figure 1

(Color online) Phase diagram in the $\left(v_0,{\rho }_0\right)$ plane. At $v_0=0$ the system exhibits a continuous mean-field transition at ${\rho }_0={\rho }_c$ from an isotropic (i) to a homogeneous polarized (HP) state. The isotropic phase survives at finite $v_0$ in the region $\rho <{\rho }_c$ bounded by the vertical dotted line (blue online). For ${\rho }_0>{\rho }_c$ there is a critical $v_c\left({\rho }_0\right)$ separating a polarized moving state with large anomalous fluctuations, named the fluctuating flocking state, at low self-propulsion speed from a high-speed phase of traveling stripes. The circles denote the values of $v_c\left({\rho }_0\right)$ obtained numerically with the error bars indicating the step size used in the computation. The dashed-dotted line (purple online) is the longitudinal instability boundary $v_{c1}^L\left({\rho }_0\right)$ obtained in Sec. 3. The dashed line (red online) is the splay instability boundary $v_c^S\left({\rho }_0\right)$, given by Eq. (A11). The top right panel is a real space snapshot of the density profile in the striped phase. The stripes travel in the direction of the white arrow that also denotes the direction of mean polarization in the high-density regions. The bottom right panel shows a real space snapshot of the density profile in the coarsening transient leading to the fluctuating flocking state at $v_0<v_c$. Density values from low (dark in grey-scale/blue online) to high (light in grey-scale/yellow online) are indicated in the side bar. The value of the density in the blue stripes is well below the critical value ${\rho }_c=0.5$, while the red stripes are well into the polarized phase.

Figure 2

(Color online) The figure displays the linear stability boundaries in the $\left(v_0,{\rho }_0\right)$ plane. All lines have been calculated using the microscopic parameter values of Table . The vertical dotted line (blue online) is the mean-field continuous transition from the isotropic (i) to the HP state. The dashed-dotted lines (purple online) are the boundaries [calculated by numerical solution of $D_{eff}\left(v_0,{\rho }_0\right)=0$, with $D_{eff}$ given by Eq. (20)] that define the region $v_{c1}^L\le v_0\le v_{c2}^L$ where the homogeneous polarized state is unstable due to the growth of coupled density and polarization fluctuation associated with spatial gradients along the direction of mean order (longitudinal instability). The linear theory predicts that the homogeneous polar state is unstable in the ruled region to the right of the vertical mean-field transition and bounded by these two lines. The dashed line (red online) is the splay instability boundary given in Eq. (A11). It terminates at a finite value at ${\rho }_0={\rho }_c$. The linear theory predicts that splay fluctuations destabilize the polar state in the entire ruled region above the dashed (red) line. The region where the system exhibits both the longitudinal and splay instabilities is cross-hatched. The longitudinal instability boundary $v_{c1}^L\left({\rho }_0\right)$ vanishes for $v_0\to 0$ and ${\rho }_0\to {\rho }_c^{+}$, in agreement with the numerics.

Figure 3

(Color online) The magnitude squared of the orientational order parameter $⟨P^2\left(t\right)⟩$ as a function of time for an isotropic initial state $\left[⟨P^2\left(t=0\right)⟩=0\right]$ and an ordered initial state $\left[⟨P^2\left(t=0\right)⟩=1\right]$, for three system sizes and $v_0=0.1$, ${\rho }_0=0.7$, corresponding to $v_c\sim 0.42$. The three curves obtained for an isotropic initial state overlap and cannot be distinguished in the figure. Both initial states approach the same macroscopically ordered state at long time although the time scale required for the isotropic initial state to reach the asymptotic steady value is much longer.

Figure 4

(Color online) The mean magnitude of the polar order parameter as a function of system size $L$ for different values of $v_0$ and ${\rho }_0=0.7$, corresponding to $v_c\simeq 0.42$. The curves appear to saturate as the system size increases, suggesting that the system remains macroscopically ordered in the thermodynamic limit.

Figure 5

(Color online) Early time, two-point density (top panel) and order-parameter (bottom panel) correlation function for $v_0=0.1<v_c=0.42$ mean density ${\rho }_0=0.7$ and $L=1024$. The dashed horizontal line indicates the value of the correlation function at which we extract the coarsening length scale $L\left(t\right)$, shown in Fig. 6.

Figure 6

(Color online) The coarsening length $L\left(t\right)$ as a function of time for the density (circles) and order-parameter (squares) correlations for $v_0=0.1$, ${\rho }_0=0.7$, and $L=1024$. For this density $v_c\simeq 0.42$ and the system was started in a disordered state. The straight line has slope of 2. Although no single scaling exponents can be extracted for the two length scales $L\left(t\right)$, it is clear that the growth with time is faster than would be obtained for $z=2$.

Figure 7

(Color online) Plot of $C_{\delta P}\left(\mathbf{r},t\right)$ for the case when $\mathbf{r}$ is parallel to ${\widehat{p}}_0$ (black-open circles) and when $\mathbf{r}$ is perpendicular to it (red-filled circles). The anisotropy of the correlation functions begins to appear at large $r$, but larger system sizes are needed to quantify the difference.

Figure 8

(Color online) Late-time two-point correlation functions of density, $C_{\rho }\left(r\right)$, and order parameter, $C_{\delta P}\left(r\right)$, fluctuations. The data are obtained from an initial ordered configuration for ${\rho }_0=0.7$, $L=512$, and various values of $v_0$, in the region corresponding to the fluctuating flocking state. Inset (top panel) shows $\left\{1-C_{\rho }\left[r/L\left(t\right)\right]\right\}$ vs the scaled distance $r/L\left(t\right)$, for various values of $v_0$. The dashed straight line shows the cuspy nature of two-point correlation function with cusp exponent $\alpha =0.6$. Inset (bottom panel) shows the scaled two-point order-parameter correlation function, $C_{\delta P}\left[r/L\left(t\right)\right]$, vs the scaled distance $r/L\left(t\right)$, for several $v_0$. The dashed straight line shows the logarithmic decay of $C_{\delta P}\left[r/L\left(t\right)\right]$.

Figure 9

(Color online) [(a)–(c)] Density and [(b)–(d)] order-parameter histograms. In panels (a) and (b) the histograms are shown for ${\rho }_0=0.7$, $v_0=0.1$, 0.3, 0.5, 0.7 and $L=128$. For small $v_0$, below the threshold value $v_c\simeq 0.42$, the histograms are unimodal (fitted by a Gaussian), indicating a uniform state. For $v_0>v_c$, the histograms are bimodal (fitted by two Gaussian curves), indicating the onset of stripes. In panels (c) and (d) the histograms are shown for $v_0>v_c$ and three values of the mean density, ${\rho }_0=0.6$, 0.7, 0.8. The position of the peaks does not change with density, while the difference in the height of the two peaks increases with increasing density.

Figure 10

(Color online) Speed of the stripes as a function of $v_0-v_c$ for ${\rho }_0=0.7$. The speed of propagation of the stripes increases linearly with $v_0$. The solid line is a guide to the eye. The dashed line is a linear fit $\sim v_0-v_c$

Figure 11

(Color online) Steady-state two-point density correlation function for three different directions with respect to the ordering direction, $90°$, $45°$, and $0°$ for $v_0>v_c$ (top panel) and for $v_0<v_c$ (bottom panel). For $v_0<v_c$, there is no directional dependence in the correlation function. For $v_0>v_c$, correlations at $90°$ to the ordering direction decay monotonically, while correlations at $0°$ and $45°$ to the ordering direction show oscillations.

Figure 12

Plot of polarization as a function of $x$ as obtained by inverting Eq. (33) for ${\rho }_0=0.7$ and $v_0=1.0$. The solution represents a propagating domain boundary between a state with $P=1$ and a state with $P=0$. For these parameters $\Lambda \sim 1.57$.

Figure 13

(Color online) The maximum growth wave vector of the longitudinal instability as a function of the self-propulsion speed $v_0$ for different densities.

Figure 14

(Color online) The maximum growth wave vector of the splay instability as a function of the self-propulsion speed $v_0$ for different densities.