Learning and investigating phenomenological models for active matter

Abstract

Active matter, such as a suspension of swimming cells or colloidal particles, or even a flock of birds, consists of self-propelled units that convert stored or ambient free energy into motion. Recent advances in high-resolution imaging techniques and particlebased simulation methods have enabled the precise characterization of the collective dynamics in such biological and engineered active systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. Motivated by these advancements, in this thesis we analytically and numerically investigate phenomenological models in the context of active fluids, and subsequently leverage recent model learning algorithms to infer continuum models directly from microscopic simulation and experimental video data.

First, we consider idealized two-dimensional swimmers with a fixed body shape (‘squirmers’) to understand the impact of yield stress on the swimming characteristics of micro-organisms. Using the Bingham constitutive law, we compute numerical flow fields around squirmers and determine their swimming speeds. Our findings demonstrate how yield stress localizes the flow and makes jet-based propulsion energetically more efficient than tangential ciliary motions. Additionally, for the related problem of non-squirming translating cylinders, we derive and provide numerical evidence for previously unestablished analytical solutions using slipline theory from ideal plasticity.

Second, we explore how a phenomenological continuum model for active suspensions can provide insights into other pattern-forming systems. Motivated in part by the complex flow patterns observed in planetary atmospheres, we investigate generalized Navier–Stokes (GNS) equations that couple nonlinear advection with a generic linear instability. This analytically tractable minimal model for fluid flows driven by internal active stresses has recently been shown to permit exact solutions on a stationary 2D sphere. Here, we extend the analysis to linearly driven flows on rotating spheres. We derive exact solutions of the GNS equations corresponding to timeindependent zonal jets and superposed westward-propagating Rossby waves, qualitatively similar to those seen in planetary atmospheres. Direct numerical simulations with large rotation rates obtain statistically stationary states close to these exact solutions. The measured phase speeds of waves in the GNS simulations agree with our analytical predictions for Rossby waves.

Finally, we address the challenge of learning macroscopic hydrodynamic equations for active matter directly from microscopic data. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through applications to a chiral active particle model mimicking swimming cells and to recent experiments of engineered active particles. In both cases, our scheme learns hydrodynamic equations that quantitatively reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data.

Overall, this thesis shows how phenomenological models for active matter can offer dynamical insights that can be further leveraged to inform data-driven equation discovery.