Large deviation techniques for model coarse graining

社会感兴趣的物理系统极其复杂。例如，与全球变暖、长期气候估计或极端气候事件预测有关的地球大气层和海洋由数量惊人的相互作用系统组成，每个系统都有许多自由度。当试图做出预测时，不可能考虑系统的全部复杂性，特别是当问题涉及长期预测或罕见事件时。实际上，多尺度系统在自然界中无处不在：与许多物理现象相关的潜在过程往往发生在截然不同的长度或时间尺度上。这打开了粗粒化和平均化的可能性，其中快速、波动的自由度可以被视为慢自由度上的有效噪波。特别是，这有助于将复杂的物理模型简化为更简单的模型，这些模型在分析或数值上是容易处理的，以使预测和对相关过程的更深入理解成为可能。稀有事件，例如相关未知数的构象变化，在这样的设置中特别有趣和丰富。在随机系统中，不太可能的波动可以将系统从其典型状态推向其他亚稳定配置，这些配置往往具有非常不同的特性。例如化学反应、相变、天气模式、蛋白质折叠或流体流动中的持久结构。在这种情况下，大偏差理论通过将自由能和熵的概念推广到任意随机系统，给出了这些构象变化的概率和机制的精确和严格的估计。在这种多尺度设置中获得显式的大偏差原理是一个巨大的挑战，因为相关的涨落源于复杂物理过程的平均，因此通常是非线性的，非高斯的，甚至是非马尔可夫的。在这种情况下，大偏差原理的计算是非常重要的，因为它将允许我们估计有效的粗粒度模型上的转移概率，而不需要考虑所有(快速的，不重要的)自由度，从而使计算变得可行。本建议关注在上述情况下理论和数值算法的发展，并使发展的技术可用于应用科学。PI将把这些用于多尺度系统和粗粒模型的大偏差方法应用于三个具体问题：(I)大气射流中的亚稳性，其中湍流涨落促进行星射流在大气流动中消失；(Ii)磁约束聚变实验，其中等离子体反应堆中边界层的构象变化阻止有效的限制；以及(Iii)光纤通信，其中光纤的随机波动导致光子通信中的比特翻转。所有理论研究工作都将导致算法或软件实现的开发，从而允许研究人员在其他与多尺度系统中罕见事件有关的领域中重复使用。

Physical systems of interest to society are extremely complex. For example, the atmosphere and ocean of Earth, relevant for questions of global warming, long time climate estimates or prediction of extreme whether events, consists of a tremendeous number of interacting systems, each with many degrees of freedom. It is impossible to consider the system in its full complexity when trying to make predictions, in particular if the questions concern long-time prediction or rare events.Indeed, multiscale systems are ubiquitous in nature: the underlying processes relevant to many physical phenomena often happen on vastly different length- or time-scales. This opens up the possibility of coarse-graining and averaging, where fast, fluctuating degrees of freedom can be considered as effective noise on slow degrees of freedom. In particular, this helps to reduce, or coarse-grain, complex physical models into much simpler models that are tractable analytically or numerically to enable prediction and deeper understanding of the involved processes.Rare events, for example conformational changes of the relevant unknowns, are particularly interesting and rich in such a setup. In stochastic systems, unlikely fluctuations can push the system from its typical state into other, meta-stable configurations with often vastly different properties. Examples are chemical reactions, phase transitions, weather patterns, protein folding, or persistent structures in fluid flow. In such situations, large deviation theory gives precise and rigorous estimates of the probabilities and mechanisms of these conformational changes, by generalising the notion of free energy and entropy to arbitrary stochastic systems.Obtaining explicit large deviation principles in this multiscale setup is a big challenge, since the associated fluctuations stem from averaging of complex physical processes, and therefore are generally non-linear, non-Gaussian, or even non-Markovian. The computation of large deviation principles in such a setup is of high importance, as it would allow us to estimate transition probabilities on the effective, coarse-grained model, without the need to consider all (fast, unimportant) degrees of freedom, thus making computation feasible.The proposal concerns itself with the development of theory and numerical algorithms in the above situation, and to make available the developed techniques to applied sciences. The PI will apply these large deviation methods for multiscale systems and coarse-grained models to three concrete problems: (i) Metastability in atmospheric jets, where turbulent fluctuations facilitate the disappearance of planetary jets in atmospheric flow, (ii) magnetically confined fusion experiments, where conformational changes in the boundary layer in plasma reactors prevent efficient confinement, and (iii) fibre-optics communications, where random fluctuations in optical fibres lead to bit-flips in photonic communication.All theoretical research efforts will result in the development of algorithms or software implementations permitting the re-use by researchers in other fields that are concerned with rare events in multiscale systems.