Relation between the angular dependence of interaction and the stability of homogeneous states in models of self-propelled particles

We investigate the stability of a homogeneous state of self-propelled particles based on a microscopic dynamical model in which particles obey Newton's equations of motion. We do not suppose any specific distance dependence for the interaction among particles. The stability is investigated for two typical angular dependences of the interaction, one of which includes an isotropic interaction. It is found that the effect of angular dependence on the stability is small except for the isotropic interaction. The result indicates that another interaction is necessary to stabilize the homogeneous state.

Figure 1

Illustration of idealized flocking. Wedges represent the positions and the directions of motion of particles. In the homogeneous state, particles are supposed to be at the sites of a triangular lattice, and to move at the same velocity. In the illustration, all particles move rightward.

Figure 2

Graphs of two functions (a) $h_1\left(\theta \right)$ and (b) $h_2\left(\theta \right)$, where $\theta$ is the angle between ${\vec{e}}_j$ and ${\vec{n}}_{\alpha j}$. ${\vec{e}}_j$ and ${\vec{n}}_{\alpha j}$ represent the direction of desired velocity of the $j\mathrm{th}$ particle and the direction from the $j\mathrm{th}$ particle to the $\alpha \mathrm{th}$ particle, respectively.

Figure 3

The stable region in the case of $h_1\left(\theta \right)$. In the shaded area, the homogeneous state is stable. Thick, thin, and dashed lines represent stability conditions (14), (40), and (19), respectively.

Figure 4

The stable region in the case of $h_2\left(\theta \right)$ with $c=0$. In the shaded area, the homogeneous state is stable. Thick, thin, and dashed lines represent stability conditions (20), (47), and (15), respectively.

Figure 5

Boundaries of the stability region expressed by (a) Eq. (14) and (b) Eq. (15) with $c=0$. In each graph, the thick solid line represents the boundary of the first condition of Eq. (14) or (15). Solid, dashed, dotted, and dashed-dotted lines represent the second ones for $a^2=1$, 3, 5, and 10, respectively.

Figure 6

Boundaries of the stability region expressed by (a) Eq. (19) and (b) Eq. (20) with $c=0$. In each graph, the thick solid line represents the boundary of the first condition of Eq. (19) or (20). Solid, dashed, and dotted lines represent the second ones for $a^2=1$, 3, and 5, respectively.

Figure 7

New stability conditions for the general case. (a) A shaded area represents the stable region defined by Eqs. (40) and (41) in the case of $h_1\left(\theta \right)$ with $a^2=3.0$. (b) A shaded area represents the stable region defined by Eq. (47) in the case of $h_2\left(\theta \right)$ with $c=0$. Dashed lines are reference lines defined by Eq. (21).

Figure 8

Positions and indices of neighbor particles.