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Left and right columns: snapshots of exact coherent structures (ECS) in active nematic channel flow

March 2022 update 
Our poster was featured in the APS 2022 Gallery of Soft Matter! See here for more information. We are poster #P0031, titled "Mosaics of active turbulence". 

What is active matter?

Much of my research focuses on active matter, which is a class of nonequilibrium materials in which energy is continually injected at the level of individual components. Such systems are important in biology and also constitute a testing bed for new ideas in nonequilibrium statistical mechanics.

In this spirit, my Ph.D. work focused on nonequilibrium steady states in systems of active colloids. The methods I employed ranged from particle-level kinetic descriptions to coarse-grained continuum hydrodynamics. Click here to learn more about active matter.

Current (postdoctoral) work

My current work also deals with active matter but focuses on time-dependent phenomena in continuum hydrodynamic models, particularly the low Reynolds number turbulence known to arise in certain active matter systems.

Active nematic channel flow

Our model system is a 2D active nematic liquid crystal, described by deterministic hydrodynamic equations for the nematic alignment tensor Q and velocity v. Under confinement, this model exhibits rich pattern formation as well as spatiotemporal chaos for large active driving. As a starting point, we have focused on an active nematic confined in a 2D channel, with periodic boundary conditions in the horizontal direction and confining walls above and below. The attracting states in this geometry are highly diverse, encompassing steady, "self-pumping" flows; complex space- and time-periodic patterns; and turbulence [Shendruk, et al (2017)].

Our goal has been to understand these phenomena using tools from dynamical systems theory and classical fluid dynamics. The details of the framework are described in the section below; broadly speaking, we develop a reduced-order, predictive model by breaking up the flow dynamics into basic components known as Exact Coherent Structures (ECS). The ECS are exact and (typically) unstable solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure, such as unstable equilibria, periodic orbits, and traveling waves. Examples are shown to the left and right as snapshots of the velocity field, where the color gradient represents the vorticity. (If viewing on a mobile device, these images will be at the bottom of the page.)

 

Though the ECS are usually dynamically unstable, a typical turbulent time evolution may linger in the vicinity of an ECS for some time before transitioning to a different ECS and continuing the process. If this is the case, then one can construct a fully nonlinear but reduced-order model for the flow dynamics, in which a typical time evolution is decomposed into a sequence of transitions within a finite set of known ECS. Recently, we demonstrated the viability of this decomposition for pre-turbulent active nematic channel flow [Wagner, et al. Phys. Rev. Lett. (2022)].

 

In contrast with statistical theories of turbulence, this approach is deterministic and exact and therefore well-suited to describe phenomena such as transitional turbulence and intermittency. Among many goals, our ongoing work seeks to connect the framework with experiments on confined active nematics.

Detailed framework

More broadly, my current research can be framed as a deterministic, dynamical systems approach to time-dependent phenomena in classical field theories, with an emphasis on turbulence phenomena.

 

The starting point is a deterministic dynamical field theory with a nonlinear evolution equation (if there are fluctuations, a mean-field description is assumed). The time evolution of the fields is envisioned as a deterministic trajectory moving in an infinite dimensional phase space. It is known from the dynamical systems literature that low-dimensional chaotic systems evolve in a phase space populated by exact invariant solutions such as equilibria and periodic orbits [see, e.g. Auerbach (1987)]. Extending this notion to physical field theories, one postulates that the corresponding infinite-dimensional phase space is populated by low-dimensional, non-chaotic, invariant sets, such as equilibria, traveling waves, periodic orbits, and invariant tori, collectively called "exact coherent structures" (ECS).

 

In turbulent systems, all ECS are dynamically unstable. Nevertheless, a typical turbulent time evolution appears as a chaotic trajectory meandering through the phase space and visiting the neighborhoods of the ECS in a recurring fashion. The transitions between ECS can be described by invariant manifolds connecting them. Because individual ECS may be far apart in phase space, these manifolds capture the global phase space geometry in a way that locally linearized descriptions cannot. Therefore, the collection of ECS and invariant manifolds constitute an exact and fully nonlinear template for the complicated spatiotemporal dynamics.

 

Being deterministic and exact, this framework addresses phenomena not easily accessible in a statistical description, such as pre-turbulent pattern formation, transitional turbulence, and intermittency. The approach has been successfully applied to classical fluids described by the Navier-Stokes equations. As mentioned above, I and others have also begun to apply the framework to field theories of complex fluids, such as active matter. While the conceptual foundations are not fundamentally new, the long-term potential of this research program is just beginning to be realized in soft matter. Key driving factors include new experimental realizations of nonequilibrium materials, accompanied by interest in optimal control methods for manipulating them (for instance, to design emergent functionality). As I suggest below, there are also open questions of fundamental interest to theoretical physics, which makes me suspect that the dynamical systems approach will attract interest in areas of physics as well.

Some open questions
  1. Turbulence in diverse physical contexts.---A working definition of turbulence is spatiotemporal chaos. However, this classification is very broad. In what ways do turbulent phenomena differ across various types of field theories? The dynamical systems approach can provide insight here where statistical methods cannot, particularly in relation to transitional turbulence and intermittent phenomena.

  2. Topological features.---At sufficiently large driving, 2D active nematics spontaneously nucleate motile defects. Turbulence is characterized by chaotic motion of these defects; however, in pre-turbulent flows they may also perform time-periodic braiding motions. The connections of the latter behavior with topological entropy are being explored [Tan, et al. Nature Physics (2019)], but an ECS-based approach is currently lacking. More generally, it would be interesting to explore topological aspects of generic classical field theories from this perspective. 3D systems in particular exhibit disclination loops with rich dynamics, and are currently not well understood.

  3. New physical systems.---In some physical contexts, an ECS-based approach has (to my knowledge) not yet been tried. It would be interesting to perform entry level calculations in such systems. Promising candidates include magnetohydrodynamics, relativistic hydrodynamics, nonequilibrium critical phenomena, and semiclassical limits of QFTs. (The last one is rather speculative, but potentially of deep fundamental interest.)

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