人群动力学建模：多尺度测量理论方法
Modelling Crowd Dynamics: a Multiscale, Measure-theoretical Approach
摘要：
我们提出了一个策略，能够描述人群的动态的基本特征。从两个角度考虑群众的行为。我们研究了人群的大规模行为，以及发生在单个行人层面的现象。我们统一微观和宏观在一个单一的模型，通过一般的质量措施和他们的运输工作。我们改进现有的建模耦合措施的混合物理论的基本思想制定的措施的理论框架。这种策略允许我们定义人群的几个组成部分，每个组成部分都有自己的偏速度。因此，我们可以研究具有不同特征的亚群之间的相互作用。我们给这些行人的微观（离散）部分所代表的特殊功能。在现实生活中，他们将扮演领导者，捕食者等的角色，因为我们感兴趣的是群体中其他人的全球行为，我们将这部分建模为一个连续体。通过确定一个合适的熵的概念，我们得到了一个熵不等式，并表明我们的模型符合克劳修斯-迪昂不等式。从这个不等式自然限制建议的速度场遵循;遵守这些限制使我们的模型与热力学兼容。我们证明了一般质量措施的时间离散运输问题的解的存在性和唯一性。此外，我们还证明了解的正性和质量守恒等性质。我们的结果是有效的质量措施，在其最一般的外观。我们给出了一个强大的计划来近似的解决方案，并说明数值双尺度微观宏观行为。我们尝试了一些场景，以捕捉紧急定性行为。最后，我们制定开放的问题和基本的研究问题，我们的建模，分析和仿真结果的启发。
英文摘要：
We present a strategy capable of describing basic features of the dynamics of crowds. The behaviour of the crowd is considered from a twofold perspective. We examine both the large scale behaviour of the crowd, and phenomena happening at the individual pedestrian's level. We unify micro and macro in a single model, by working with general mass measures and their transport. We improve existing modelling by coupling a measure-theoretical framework with basic ideas of mixture theory formulated in terms of measures. This strategy allows us to define several constituents of the crowd, each having its own partial velocity. We can thus examine the interaction between subpopulations that have distinct characteristics. We give special features to those pedestrians that are represented by the microscopic (discrete) part. In real life they would play the role of leaders, predators etc. Since we are interested in the global behaviour of the rest of the crowd, we model this part as a continuum. By identifying a suitable concept of entropy, we derive an entropy inequality and show that our model agrees with a Clausius-Duhem-like inequality. From this inequality natural restrictions on the proposed velocity fields follow; obeying these restrictions makes our model compatible with thermodynamics. We prove existence and uniqueness of a solution to a time-discrete transport problem for general mass measures. Moreover, we show properties like positivity of the solution and conservation of mass. Our results are valid for mass measures in their most general appearance. We give a robust scheme to approximate the solution and illustrate numerically two-scale micro-macro behaviour. We experiment with a number of scenarios, in order to capture the emergent qualitative behaviour. Finally, we formulate open problems and basic research questions, inspired by our modelling, analysis and simulation results.
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