We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, \(\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)\), with anticipation increment \(\tau >0\). As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected to emerge into flocking due to alignment of velocities, and spatial concentration due to confining potentials. We treat both the discrete dynamics and large crowd hydrodynamics, proving the decisive role of anticipation in driving such systems with attractive potentials into velocity alignment and spatial concentration. We also study the concentration effect near equilibrium for anticipated-based dynamics of pair of agents governed by attractive–repulsive potentials.

我们研究了由径向势驱动的系统的长时间行为，这些行为对预期位置做出反应，\（\ mathbf {x} ^ \ tau（t）= \ mathbf {x}（t）+ \ tau \ mathbf {v} （t）\），预期增量为\（\ tau> 0 \）。作为一种特殊情况，此类系统产生了著名的Cucker-Smale对准模型，以及成对相互作用。从这个角度来看，由于速度和空间集中度的原因，这种预期驱动的系统有望蜂拥而至。由于局限的潜力。我们同时处理离散动力学和大群流体动力学，证明了预期在驱动具有吸引力的系统（如速度对齐和空间集中）中起决定性作用。我们还研究了由吸引-排斥势控制的一对试剂的基于预期的动力学在平衡附近的浓度效应。