论文标题
环境平均
Environmental Averaging
论文作者
论文摘要
许多自组织动力学模型的经典示例，包括Cucker-Smale，Motsch-Tadmor，Multipecies等，包括基于密度加权平均方案方案的对齐力。这些协议可以被视为“环境平均”的特殊情况。在本文中，我们将此概念形式化，并引入了一个统一的框架，以进行对齐模型的系统分析。 介绍了一系列研究，包括确定性和随机设置的平均场限制，单基因和麦克斯韦政权中的流体动力限制，耗散动力学模型的低调和全球放松，这是基于链连接性和光谱差距分析的几个一般对齐结果。这些研究涵盖了许多已知结果并揭示了新的结果，其中包括基于连接性条件的渐近一致性标准，对对齐力的光谱差距的新估计不依赖于宏观密度的上限，基于平稳环境均衡的Fokker-Planck-Planck-Planck-Planck-Planck-Plangk-Planck-Planck-Planck-Planck-Planck-Planck-plancker-PlanciTivitive。结果，我们为Fokker-Planck-Alignment模型的全球解决方案建立了无条件的放松结果，该模型对先前已知的扰动结果提供了实质性改善。
Many classical examples of models of self-organized dynamics, including the Cucker-Smale, Motsch-Tadmor, multi-species, and several others, include an alignment force that is based upon density-weighted averaging protocol. Those protocols can be viewed as special cases of `environmental averaging'. In this paper we formalize this concept and introduce a unified framework for systematic analysis of alignment models. A series of studies are presented including the mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, hypocoercivity and global relaxation for dissipative kinetic models, several general alignment results based on chain connectivity and spectral gap analysis. These studies cover many of the known results and reveal new ones, which include asymptotic alignment criteria based on connectivity conditions, new estimates on the spectral gap of the alignment force that do not rely on the upper bound of the macroscopic density, uniform gain of positivity for solutions of the Fokker-Planck-Alignment model based on smooth environmental averaging. As a consequence, we establish unconditional relaxation result for global solutions to the Fokker-Planck-Alignment model, which presents a substantial improvement over previously known perturbative results.
