Energy Cost for Flocking of Active Spins: The Cusped Dissipation Maximum at the Flocking Transition

We study the energy cost of flocking in the active Ising model (AIM) and show that, besides the energy cost for self-propelled motion, an additional energy dissipation is required to power the alignment of spins. We find that this additional alignment dissipation reaches its maximum at the flocking transition point in the form of a cusp with a discontinuous first derivative with respect to the control parameter. To understand this singular behavior, we analytically solve the two- and three-site AIM models and obtain the exact dependence of the alignment dissipation on the flocking order parameter and control parameter, which explains the cusped dissipation maximum at the flocking transition. Our results reveal a trade-off between the energy cost of the system and its performance measured by the flocking speed and sensitivity to external perturbations. This trade-off relationship provides a new perspective for understanding the dynamics of natural flocks and designing optimal artificial flocking systems.

Figure 1

(a) The average flocking speed $v$ versus the total dissipation ${\dot{W}}_{\mathrm{tot}}$ for fixed values of $E_0$ and increasing $\epsilon$. (b) The average flocking speed (blue) and alignment dissipation (black) for $\epsilon =0.3$. The red dashed line is the transition point $E_c$ above which $v>0$. The inset shows the exponential decay of dissipation at large $E_0$. $L_x=300$, $L_y=100$, $\overline{\rho }=N/(L_x{L}_y)=5$, $D=1$, and $\omega =1$. See Sec. I A in the Supplemental Material for details [33].

Figure 2

(a) The effective free energy landscape demonstrates the existence of a nonequilibrium phase transition in the two-site AIM. $N=50$, $D=\omega =1$. (b) The alignment dissipation of the two-site AIM with different finite $N$ (solid lines) and infinite $N$ (red dashed line) and the full AIM (purple dots, normalized according to the text). Inset: the curvature at the dissipation maximum in the two-site AIM.

Figure 3

The energy-speed-sensitivity trade-off in the two-site AIM. (a) Contours for constant $v$ in the $(\epsilon ,E_0)$ plane. (b) The total dissipation and sensitivity along different $v$ contours. $N=40$, $D=\omega =1$.

Figure 4

Generality of the alignment dissipation maximum at flocking transition. The heat maps show the alignment dissipation density ${\dot{w}}_a/(L_x{L}_y)$ for different combinations of (a) $(E_0,\epsilon )$ and (b) $(E_0,\omega )$. The red lines indicate the flocking transition as measured by the flocking velocity $v$. $D=1$, $L_x=300$, $L_y=100$, $\overline{\rho }=N/(L_x{L}_y)=5$. $\omega =1$ for (a); $\epsilon =0.3$ for (b). The black dashed line in (a) shows the parameter values for Fig. 1.