Flocking in $d=2$ is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

中文翻译：

---

植绒微观模型中的不可逆特征

涌入$d=2$是一种真正的非平衡现象，其中不可逆性是必不可少的组成部分。我们研究了一类最小植绒模型，其不可逆性的唯一来源是自推进，并使用熵产生率 (EPR) 来量化其相图中的平衡偏离。EPR 在有序-无序转换附近最大，其中交互网络的重新洗牌很快。我们表明，不可逆性的特征以群体微观状态的稳态分布不对称的形式出现。这些不对称性是所考虑的自推进系统中时间反演对称性破坏的结果，与交互细节无关。在度量成对力的情况下，它们减少到粒子对分布中的局部不对称。