Active matter describes a class of systems that are maintained far from equilibrium by driving forces acting on the constituent particles. Here I will focus on confined active nematics, which exhibit especially rich flow behavior, ranging from structured patterns in space and time to disordered turbulent flows. To understand this behavior, I will take a deterministic dynamical systems approach, beginning with the hydrodynamic equations for the active nematic. This approach reveals that the infinite-dimensional phase space of all possible flow configurations is populated by Exact Coherent Structures (ECS), which are exact solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure; examples include unstable equilibria, periodic orbits, and traveling waves. The ECS are connected by dynamical pathways called invariant manifolds. The main hypothesis in this approach is that turbulence corresponds to a trajectory meandering in the phase space, transitioning between ECS by traveling on the invariant manifolds. Similar approaches have been successful in characterizing high Reynolds number turbulence of passive fluids. Here, I will present the first systematic study of active nematic ECS and their invariant manifolds and discuss their role in characterizing the phenomenon of active turbulence.

The Smoluchowski equation has often been used as the starting point of many continuum models of active suspensions. However, its six-dimensional nature depending on time, space and orientation requires a huge computational cost, fundamentally limiting its use for large-scale problems, such as mixing and transport of active fluids in turbulent flows. Despite the singular nature in strain-dominant flows, the generalised Taylor dispersion (GTD) theory (Frankel & Brenner 1991, J. Fluid Mech. 230:147-181) has been understood to be one of the most promising ways to reduce the Smoluchowski equation into an advection-diffusion equation, the mean drift and diffusion tensor of which rely on ‘local’ flow information only. In this talk, we will introduce an exact transformation of the Smoluchowski equation into such an advection-diffusion equation requiring only local flow information. Based on this transformation, a new advection-diffusion equation will subsequently be proposed by taking an asymptotic analysis in the limit of small particle velocity. With several examples, it will be demonstrated that the new advection-diffusion model, non-singular in strain-dominant flows, outperforms the GTD theory.