Linking dynamics to scaling laws in physical and biological systems

将动力学与物理和生物系统中的尺度定律联系起来

• 批准号: RGPIN-2019-05443
• 负责人: vanVeen, Lennaert
• 金额: $ 1.38万
• 依托单位: University of Ontario Institute of Technology
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=765051
• 关键词: Linking; dynamics; scaling; laws; physical

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.

该发现计划包括非线性动力学和统计学交叉领域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。在每一种现象中，复杂的非线性动力学都会产生随时间、空间或实现而平均的量的鲁棒特性。在流体湍流中，相干结构的连续形成和破坏平均而言共同产生了著名的科尔莫哥洛夫幂定律，用于空间尺度上的能量分布。柯尔莫哥洛夫的理论表明，相干结构以自相似的方式相互作用，但这种相互作用的动力学本质仍不清楚。在界面生长的研究中，我们遇到了相反的问题。在 Kuramoto 和 Sivashinsky 提出的模型中，我们准确地知道预期的动态。令人惊讶的是，这些动力学导致什么统计行为是一个悬而未决的问题。三十五年多前，Yakhot 推测该模型的统计特性与界面增长的各种随机模型的统计特性相同。完全确定性的 Kuramoto-Sivashinsky 模型从根本上来说比流体湍流模型简单，但迄今为止还没有确凿的证据来支持这一猜想。我们将使用基于 GPU 的尖端计算动力系统理论实现来为这些经典问题提供新的视角，这些问题已经经历了数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。由于实验揭示了单个细胞运动的细节，因此常见的建模方法是基于代理的模拟。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。这种模拟可以展示一致移动的细胞簇的形成。然而，即使借助 GPU 计算，我们也只能模拟微观上的小簇，而在培养皿中会形成更大的菌落。面临的挑战是制定一个局部平均、连续的簇形成模型，在精神上更接近流体运动方程，而不是基于代理的模型。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，并更好地了解观察到的生物膜的形成，例如在人体内的医疗植入物上。正在考虑的问题位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。各个级别的学生都将受益于跨学科培训机会，包括现代计算技术，并为加拿大就业市场上对定量分析和复杂流程优化不断增长的需求做好准备。