Stochastic active flows and interacting particle systems

描述连续体的数学模型已经非常成功地用于解释各种系统的行为，而不仅仅是激发了它们最初发展的液体流动。例如，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。这类基于偏微分方程的模型没有捕捉到由离散移动介质组成的“流体”中有限尺寸效应引起的噪声诱导波动的影响。基于区隔的药物相互作用方法（例如，在反应扩散系统中）允许纳入随机效应并将这些效应作为一个整体对系统进行量化。当药剂之间的相互作用不局限于它们的局部附近时，这种方法就不太方便了，因为只有同一隔间中的粒子需要考虑。微观水平的模型允许对粒子之间相互作用的基本规则的最大程度的准确性。然而，对于许多系统来说，使用这些模型来理解整个系统的演化是一项艰巨的挑战。我博士学位的第一个任务将是明确地将基于微观连续体的相互作用随机主体模型与一维宏观运动学流体模型联系起来，从而能够分析由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，这些模型可以理解为成对粒子相互作用的大粒子极限。然后，在研究相互作用的具体规则和由此产生的spde之前，我将为广义系统的进化找到一个随机PDE。我将比较模型特定情况下spde的模拟与微观模拟，检查是否与总体密度演变一致，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群体行为中观察到的铣削模式形成的潜力。对于一维和二维，我将寻找双稳定的系统，它可以用粒子的成对相互作用模型来建模。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SPDE来量化密度演变。

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.