Swarms with canonical active Brownian motion

We present a swarm model of Brownian particles with harmonic interactions, where the individuals undergo canonical active Brownian motion, i.e., each Brownian particle can convert internal energy to mechanical energy of motion. We assume the existence of a single global internal energy of the system. Numerical simulations show amorphous swarming behavior as well as static configurations. Analytic understanding of the system is provided by studying stability properties of equilibria.

Figure 1

Time evolution of the deterministic equations (14, 15, 16, 17, 18, 19, 20) for $K$ (long dashed line), $U$ (solid line), $S$ (short dashed line), and $e$ (solid line) for the collapse of the swarm to the minimum of the external potential ($E_1$). Simulation parameters are $c=1$, $d_1=1/2$, $d_2=1$, $\gamma =2$, $k_1=1/4$, and $k_2=1/4$. The initial conditions are $U(0)=K(0)=1/c=1$ and $S(0)=e(0)=T(0)=V(0)=L(0)=0$. ($T$, $V$, and $L$ remain equal to zero in time and are not shown for simplicity of presentation, also in the following simulations.)

Figure 2

Relaxation to a jellylike state ($E_2$), according to Eqs. (14, 15, 16, 17, 18, 19, 20). Simulation parameters are $c=1$, $d_1=2$, $d_2=1$, $\gamma =2$, $k_1=1/2$, and $k_2=1/2$. The initial conditions are $U(0)={10}^{-4}$ and the remaining six variables equal zero.

Figure 3

State of amorphous swarming ($E_3$), according to Eqs. (14, 15, 16, 17, 18, 19, 20). Simulation parameters are $c=1/3$, $d_1=-1$, $d_2=1$, $\gamma =1$, $k_1=1/16$, and $k_2=1/128$. The initial conditions for $U$ and $K$ equal the expected equilibrium values while the remaining five variables equal zero.