Linking dynamics to scaling laws in physical and biological systems

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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.