Collective Motion of Self-Propelled Particles with Memory

We show that memory, in the form of underdamped angular dynamics, is a crucial ingredient for the collective properties of self-propelled particles. Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular velocity, we uncover a rich variety of collective phases not observed in usual overdamped systems, including vortex lattices and active foams. In a model with strictly nematic interactions the smectic arrangement of Vicsek waves giving rise to global polar order is observed. We also provide a calculation of the effective interaction between vortices in the case where a telegraphic noise process is at play, explaining thus the emergence and structure of the vortex lattices observed here and in motility assay experiments.

Figure 1

Phase diagrams of the models defined by Eqs. (1) and (2). See main text and/or Fig. 2 for definitions of the various phases. (a) Nematic model. (b) Polar model.

Figure 2

Various phases for the nematic (a)–(d) and polar (e)–(g) model. Corresponding movies are available in Ref. [28]. Top: local orientational order [nematic (a),(b); polar (c)–(g)]; color stands for orientation, intensity for modulus. Bottom: superimposed density of clockwise (CW, $\omega <0$, green) and counterclockwise (CCW, $\omega >0$, red) particles. Color maps are on the right, with ${\rho }_M=2$ for (a)–(e), 4 for (f), and 5 for (g). Parameters (${\rho }_0$, $\tau$): (a) (1,100), (b) (1,40), (c) (0.56,10), (d) (1.78,20), (e) (1,100), (f) (4.22, 17.5), and (g) (3.16, 15). Periodic square of linear size 512, $\alpha =0.1$, $⟨{\omega }^2{⟩}^{1/2}=0.03$ (0.06) for the nematic (polar) model.

Figure 3

Space-time diagram in the laning phase (nematic model). Rectangular domain $(L_x,L_y)=(1024\times 256)$ with global nematic order along $y$ (periodic boundary conditions). Local polar order averaged over $y$ is represented (color map on right). Parameters: $({\rho }_0,\tau )=(1.33,20)$ (other parameters as in Fig. 2 for the nematic model).

Figure 4

Continuous description for vortex lattice formation. Top (bottom) row: polar (nematic) model. (a),(b) Effective interaction. (c),(d) Dispersion relation (eigenvalue for $F_{+}-F_{-}$, $F_{+}+F_{-}$). (e),(f) Steady states when ${\rho }_0\tau$ is larger than the critical value. (e) shows $F_{+}-F_{-}$, and (f) shows $F_{+}+F_{-}$. Periodic square of linear size 40, $D=10$, $\alpha =0.05$, and ${\rho }_0=0.02$. In the polar (nematic) interaction case, ${\rho }_0\tau =0.2$ (0.17).