Stochastic active flows and interacting particle systems

随机主动流和相互作用的粒子系统

• 批准号: 2284235
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2284235
• 关键词: Stochastic; active; flows; interacting; particle

Mathematical models describing continua have been used very successfully to explain the behaviour of a wide rangeof systems beyond the liquid flows that inspired their initial development. For example, the formation of traffic jamsand the collective behaviour of large animal groups. However, these models typically take a purely macroscopicapproach to explain these phenomena. This class of PDE-based models does not capture the effects of noiseinducedfluctuations arising from finite size effects in 'fluids' composed of discrete moving agents. Compartmentbased-methods for the interaction of agents (in reaction diffusion systems for instance) allow for the incorporation ofstochastic effects and quantification of these effects on the system as a whole. This method is less convenient wheninteractions between agents are not confined to their local vicinity where only the particles in the same compartmentneed to be considered. Microscopic level models allow for the greatest level of accuracy to the underlying rules ofinteractions between particles. However, using these models to understand the evolution of the system as a whole isa difficult challenge for many systems.The first task of my PhD will be to explicitly link microscopic continuum based models of interacting stochastic agentswith macroscopic kinematic fluid models in one dimension, enabling an analysis of the effects of noise induced byfinite size effects. This will require the classification of the types of macroscopic models which can be understood asthe large particle limit of pair-wise particle interactions. I will then find a stochastic PDE for the evolution of thegeneralised system before looking at specific rules of interactions and the SPDEs which result from this. I willcompare simulations of the SPDEs for specific cases of the model with the microscopic simulations, checking foragreement with the overall density evolution and quantifying the scale of the stochastic effects.Secondly, I will look into the case of two dimensions. Initially focusing on conservative interactions between particles,I will then look at the potential for milling pattern formation observed in many swarming behaviours. For both one andtwo dimensions I will look for bi-stable systems which could be modelled using pair-wise interaction models ofparticles. The mean switching time between these two states, observed in cases such as locust swarming, could thenbe quantified from the SPDE for the density evolution.

描述连续体的数学模型已被非常成功地用来解释除液体流动之外的各种系统的行为，这激发了它们的最初发展。比如交通堵塞的形成和大型动物群体的集体行为。然而，这些模型通常采用纯粹的宏观方法来解释这些现象。此类基于偏微分方程的模型无法捕获由离散移动主体组成的“流体”中的有限尺寸效应引起的噪声引起的波动的影响。基于隔室的试剂相互作用方法（例如在反应扩散系统中）允许纳入随机效应并量化这些效应对整个系统的影响。当代理之间的相互作用不限于其局部附近（仅需要考虑同一隔室中的粒子）时，此方法不太方便。微观模型可以最大程度地准确描述粒子之间相互作用的基本规则。然而，使用这些模型来理解整个系统的演化对于许多系统来说是一个艰巨的挑战。我的博士学位的第一个任务是将基于微观连续体的相互作用随机代理模型与一维宏观运动流体模型明确联系起来，从而能够分析有限尺寸效应引起的噪声影响。这将需要对宏观模型的类型进行分类，宏观模型可以理解为成对粒子相互作用的大粒子极限。然后，在研究特定的相互作用规则和由此产生的 SPDE 之前，我将找到广义系统演化的随机偏微分方程。我将比较模型特定情况下的 SPDE 模拟与微观模拟，检查与整体密度演化的一致性并量化随机效应的规模。其次，我将研究二维的情况。首先关注粒子之间的保守相互作用，然后我将研究在许多集群行为中观察到的研磨模式形成的潜力。对于一维和二维，我将寻找可以使用粒子的成对相互作用模型进行建模的双稳态系统。在蝗虫群等情况下观察到的这两种状态之间的平均切换时间可以通过密度演化的 SPDE 进行量化。