Self-propelled rods are a facet of the field of active matter relevant to many physical systems ranging in scale from shaken granular media and bacterial alignment to the flocking dynamics of animals. In this paper we develop a model for nematic alignment of self-propelled rods interacting through binary collisions. We avoid phenomenological descriptions of rod interaction in favor of rigorously using a set of microscopic-level rules. Under the assumption that each collision results in a small change to a rod's orientation, we derive the Fokker-Planck equation for the evolution of the kinetic density function. Using analytical and numerical methods, we study the emergence of the nematic order from a homogeneous, uniform steady state of the mean-field equation. We compare the level of orientational noise needed to destabilize this nematic order and compare our results to an existing phenomenological model that does not explicitly account for the physical collisions of rods. We show the presence of an additional geometric factor in our equations reflecting a reduced collision rate between nearly aligned rods that reduces the level of noise at which nematic order is destroyed, suggesting that alignment that depends on purely physical collisions is less robust.

中文翻译：

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自走杆向列对齐的平均场模型

自推进杆是与许多物理系统相关的活性物质领域的一个方面，其规模从摇动的颗粒介质和细菌排列到动物的植绒动力学。在本文中，我们开发了一种通过二元碰撞相互作用的自推进杆向列对齐模型。我们避免对杆相互作用的现象学描述，而是严格使用一组微观级别的规则。在每次碰撞都会导致杆方向发生微小变化的假设下，我们推导出动力学密度函数演化的 Fokker-Planck 方程。使用解析和数值方法，我们研究了从平均场方程的均匀、均匀稳态中出现的向列顺序。我们比较了破坏这种向列顺序所需的定向噪声水平，并将我们的结果与现有的现象学模型进行比较，该模型没有明确解释棒的物理碰撞。我们在方程中显示了一个额外的几何因子的存在，反映了几乎对齐的棒之间的碰撞率降低，这降低了向列顺序被破坏的噪声水平，这表明依赖于纯物理碰撞的对齐不太稳健。