Linking dynamics to scaling laws in physical and biological systems

这个发现项目包括非线性动力学和统计学交叉的三个相互关联的研究方向。我将考虑流体湍流、界面生长和细胞的运动性。在这些现象中，复杂的非线性动力学产生了随时间、空间或实现平均的数量的鲁棒性。在流体湍流中，连贯结构的连续形成和破裂共同产生了著名的柯尔莫哥洛夫能量在空间尺度上分布的幂律。Kolmogorov的理论表明，相干结构以一种自相似的方式相互作用，但这种相互作用的动力学性质仍然不为人所知。在研究界面生长时，我们遇到了相反的问题。在Kuramoto和Sivashinsky的模型中，我们确切地知道会发生什么动态变化。令人惊讶的是，这些动态导致的统计行为是一个悬而未决的问题。35年前，Yakhot推测该模型的统计性质与大量界面生长随机模型的统计性质相同。完全确定的Kuramoto-Sivashinsky模型从根本上来说比流体湍流模型更简单，但迄今为止还没有确凿的证据支持这一猜想。我们将使用尖端的，基于gpu的计算动力系统理论实现来揭示这些经典问题的新亮点，这些问题已经经受了几十年的理论和数值研究。细胞运动的数学描述比流体湍流和界面形成的数学描述要年轻得多。由于实验已经揭示了单个细胞运动的细节，一种常见的建模方法是基于主体的模拟。在这种方法中，人们模拟单个细胞及其相互作用的方式，例如通过碰撞和对齐。这样的模拟可以展示细胞群的形成，这些细胞群的运动是一致的。然而，即使在GPU计算的帮助下，我们也只能模拟微观上的小集群，而在培养皿中形成了更大的菌落。难点在于如何建立一个局部平均的、连续的簇形成模型，在精神上更接近流体运动方程，而不是基于智能体的模型。对这种连续模型的研究将使我们能够预测集体运动的宏观特性，并更好地理解所观察到的生物膜的形成，例如，在人体内的医疗植入物上。所考虑的问题处于连续介质力学研究的前沿，需要流体物理学、动力系统理论和科学计算的创新结合来回答。各个层次的学生都将受益于跨学科的培训机会，包括现代计算技术，并为加拿大就业市场对定量分析和复杂流程优化日益增长的需求做好准备。

This Discovery program comprises three interrelated directions of research in the intersection of nonlinear dynamics and statistics. I will consider fluid turbulence, interface growth and the motility of cells. In each of these phenomena, intricate nonlinear dynamics give rise to robust properties of quantities averaged over time, space or realizations. In fluid turbulence, the continuous formation and breakdown of coherent structures conspire to produce, on average, the famous Kolmogorov power law for the distribution of energy over spatial scales. Kolmogorov's theory suggests that the coherent structures interact in a self-similar fashion, but the dynamical nature of such interaction remains ill-understood. In the study of interface growth, we encounter the opposite problem. In a model due to Kuramoto and Sivashinsky, we know precisely what dynamics to expect. Surprisingly, it is an open question what statistical behaviour these dynamics result in. Over thirty-five years ago, Yakhot conjectured that the statistical properties of the model are the same as those of a wide class of stochastic models of interface growth. The entirely deterministic Kuramoto-Sivashinsky model is fundamentally simpler than that of fluid turbulence, yet no conclusive evidence to support the conjecture has been produced to date. We will use cutting-edge, GPU-based implementations of computational dynamical systems theory to shed new light on these classical problems, that have withstood decades of theoretical and numerical study. The mathematical description of cell motility is much younger than that of fluid turbulence and interface formation. Since experiments have revealed details of individual cell motion, a common modelling approach is agent-based simulation. In this approach, one simulates individual cells and the way they interact, for instance by colliding and aligning. Such simulations can exhibit the formation of clusters of cells that move in unison. However, even with the aid of GPU computing, we can only simulate microscopically small clusters, while in a Petri dish much larger colonies are formed. The challenge is to formulate a locally averaged, continuous model of cluster formation, closer in spirit to equations for fluid motion than to agent-based models. The study of such continuous models will allow us to predict macroscopic properties of collective motion and better understand the formation of biofilms observed, for instance, on medical implants inside the human body. The questions under consideration lie at the forefront of research in continuum mechanics and will require an innovative mixture of fluid physics, dynamical systems theory and scientific computing to answer. Students on all levels will benefit from the interdisciplinary training opportunities, including modern computational techniques, and be prepared for the ever growing demand for quantitative analysis and optimization of complex processes on the Canadian job market.