Stochastic active flows and interacting particle systems

描述连续介质的数学模型已非常成功地用于解释除了激发其最初发展的液体流动之外的各种系统的行为。例如，交通堵塞的形成和大型动物群体的集体行为。然而，这些模型通常采取纯粹的宏观方法来解释这些现象。这类偏微分方程为基础的模型不捕捉noiseinducedfluctuations所产生的有限尺寸效应的离散移动代理组成的“流体”的影响。隔室为基础的方法的相互作用的代理（在反应扩散系统，例如）允许纳入ofstochastic效应和量化这些影响的系统作为一个整体。这种方法是不太方便时，代理之间的相互作用并不局限于他们的本地附近，只有在同一compartmentneed被考虑的粒子。微观层次的模型允许最大程度的准确性粒子之间的相互作用的基本规则。然而，使用这些模型来理解系统的演化作为一个整体伊萨个困难的挑战，许多systems.The第一个任务，我的博士将明确连接微观连续的相互作用随机agentsbased模型与宏观运动流体模型在一个维度上，使有限尺寸效应引起的噪声的影响进行分析。这就需要对宏观模型的类型进行分类，宏观模型可以被理解为成对粒子相互作用的大粒子极限。然后，我将找到一个随机PDE的发展thegeneralized系统之前，在特定的规则的相互作用和SPDE的结果，从这一点。我将比较模拟的SPDE的具体情况下的模型与微观模拟，检查foragreement与整体密度演变和量化的规模的随机效应。首先关注粒子之间的保守相互作用，然后研究在许多群集行为中观察到的铣削图案形成的可能性。对于一维和二维，我将寻找双稳态系统，可以使用粒子的成对相互作用模型建模.这两种状态之间的平均切换时间，观察到的情况下，如蝗虫成群，然后可以量化的密度演化的SPDE。

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.