In particle systems, flocking refers to the phenomenon where particles' individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behavior. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. In this paper, we introduce a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations.

翻译：在粒子系统中，集群现象指粒子的个体速度最终趋于一致。Cucker-Smale模型是描述该行为的经典数学框架。现有大多数Cucker-Smale模型的连续描述采用以粒子位置和速度为独立变量的偏微分方程，从而完整描述了粒子平均场极限动力学。本文提出一种新颖的简化惯性PDE模型，该模型仅包含两个依赖于粒子位置的方程。与其他简化模型不同，我们的模型并非从平均场极限推导而来，而是直接在经验密度层面进行模型简化，从而能够与底层粒子动力学建立直接联系。我们对简化模型进行了全面的理论分析，证明：首先，该简化PDE满足直观可解释的连续集群定义；其次，在特定情形下可以完整量化PDE解与粒子系统之间的差异。理论结果均通过数值模拟验证。