The paper treats an agent-based model with averaging dynamics to which we refer as the K-averaging model. Broadly speaking, our model can be added to the growing list of dynamics exhibiting self-organization such as the well-known Vicsek-type models (Aldana et al. in: Phys Rev Lett 98(9):095702, 2007; Aldana and Huepe in: J Stat Phys 112(1–2):135–153, 2003; Pimentel in: Phys. Rev. E 77(6):061138, 2008). In the K-averaging model, each of the N particles updates their position by averaging over K randomly selected particles with additional noise. To make the K-averaging dynamics more tractable, we first establish a propagation of chaos type result in the limit of infinite particle number (i.e. \(N \rightarrow \infty \)) using a martingale technique. Then, we prove the convergence of the limit equation toward a suitable Gaussian distribution in the sense of Wasserstein distance as well as relative entropy. We provide additional numerical simulations to illustrate both results.

本文对待与平均动态的基于代理的模式，这是我们参考的ķ -平均模型。从广义上讲，我们的模型可以添加到越来越多的展示自组织的动力学列表中，例如著名的 Vicsek 型模型（Aldana 等人在：Phys Rev Lett 98(9):095702, 2007; Aldana and Huepe在：J Stat Phys 112(1-2):135-153, 2003; Pimentel 在：Phys. Rev. E 77(6):061138, 2008)。在K平均模型中，N 个粒子中的每一个都通过对带有附加噪声的K 个随机选择的粒子进行平均来更新它们的位置。为了使K- 平均动力学更易于处理，我们首先使用鞅技术建立了一个在无限粒子数（即\(N \rightarrow \infty \)）的极限下的混沌类型的传播。然后，我们证明了极限方程在 Wasserstein 距离和相对熵意义上向合适的高斯分布收敛。我们提供了额外的数值模拟来说明这两个结果。