Computational and theoretical study of parameter scaling in particle systems

基于主体的模型用于模拟许多交互主体的行为，例如昆虫，鱼类，机器人或人类。一般来说，代理具有共同的交互规则。 对于这些基于代理的模型，总是存在如何选择参数以便最好地预测被建模的群体的行为的问题。一些模型，如Czirok-Vicsek模型，最初是在离散时间内制定的，而其他模型，如Cucker-Smale模型，是在连续时间内制定的。生物学和社会科学中的许多基于主体的模型使用前向欧拉方法来模拟模型。 因此，即使模型最初是在连续时间内制定的，时间步长的选择往往成为模拟中的一个参数。 除了时间步长之外，还必须选择药剂数量、颗粒密度和任何适用的相互作用范围。 但是当这些参数改变时会发生什么呢？为了保持相同的动力学，其他参数是否应该随着粒子密度的变化而变化？该项目的目标是了解离散时间基于代理的模型中的参数，包括粒子数量，时间步长和相互作用范围等参数，应该如何相互缩放，以便在不同的参数值下表现出相同的大规模动态。 另一个感兴趣的问题是在连续极限中会发生什么，即当时间步长为零并且代理的数量为无穷大时。 这些限制如何影响模型的动态？如果时间允许，本项目的进一步目标是探索这些模型的应用和相关PDE模型的推导的影响，这项研究将对目前采用基于代理的模型的许多研究领域产生广泛的影响。 这些模型经常应用于生物学、物理学和社会科学。 在这些领域中，被建模的系统通常包括大量的生物体，如人、鱼、昆虫、鸟或机器人，它们之间相互作用。 因此，通常有必要通过基于代理的模型来近似种群的行为，其中每个代理代表一组个体，即使交互规则是根据个体生物之间的相互作用推导出来的。 因此，如何缩放的参数作为模拟中的代理的数量变化的问题是这些模型的成功应用的核心。此外，在应用这些基于代理的模型的地方，离散时间版本经常被用来代替连续动态系统，并且很容易忽略时间步长应该如何与其他参数缩放的问题。因此，必须理解和传播这些缩放的效果，因为它们可以显著改变模型的动态。 通过这种方式，数学、物理、生物和社会科学都可以从PI及其合作者的调查中获得重要信息。

Agent-based models are used to simulate the behavior of many interacting agents, such as insects, fish, robots, or humans. Generally, agents are taken to have common interaction rules. With these agent-based models, there is always the question of how to choose parameters in order to best predict the behavior of the population being modeled. Some models, such as the Czirok-Vicsek model, are originally formulated in discrete time, while others, such as the Cucker-Smale model, are formulated in continuous time. Many agent-based models in biology and the social sciences use the Forward Euler Method to simulate the model. Hence, even if the model is originally formulated in continuous time, the choice of timestep often becomes a parameter in the simulations. In addition to the timestep, the number of agents, particle density, and any applicable ranges of interaction must be chosen. But what happens as these parameters are changed? Should the other parameters change as the particle density changes in order to maintain the same dynamics? The goal of this project is to understand how the parameters in discrete-time agent-based models, including the number of particles, the timestep, and parameters such as the interaction range, should scale with one another in order exhibit the same large-scale dynamics with different parameter values. Another question of interest is what happens in the continuum limit, i.e. when the timestep is taken to zero and the number of agents is taken to infinity. How do such limits affect the dynamics of the model? A further goal of this project, if time permits, is to explore the impacts of this investigation on applications of these models and on the derivation of associated PDE models.This research stands to have a broad impact on many research areas which currently employ agent-based models. These models are often applied in biology, physics, and social science. In these fields, the system being modeled generally includes a prohibitively large number of organisms, such as people, fish, insects, birds, or robots, interacting among themselves. As such, it is often necessary to approximate the behavior of the population by an agent-based model where each agent represents a group of individuals, even though rules of interaction are derived with interactions among individual organisms in mind. Thus, the question of how to scale the parameters as the number of agents in the simulation changes is central to the successful application of these models. Furthermore, where these agent-based models are being applied, discrete-time versions are often employed in place of continuous dynamical systems, and it is easy to ignore the question of how the timestep should scale with other parameters. Hence, it is imperative that the effect of these scalings be understood and disseminated, since they can significantly change the dynamics of the model. In this way, mathematics, physics, biology, and social science all stand to gain essential information from the investigations of the PI and her collaborators.