Anticipation breeds alignment

Understand

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 anticipated-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.