The nonequilibrium thermodynamics of flocking of active spins
The collective coherent motion known as flocking is a nonequilibrium phenomenon and is sustained by a continuous input of free energy. By studying the energy dissipation of the active Ising model (AIM), we show that the energy cost can be decomposed into two parts, namely the cost for self-propelled motion and an additional energy dissipation required to align spins in order to maintain the flocking order. 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 interaction strength. To understand this singular behavior, we analytically solve the two-site 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.