Session N29: Non-Reciprocity in Soft and Active Matter II

Odd mechanics of active slender structures

Living systems are chiral on multiple scales, from constituent biopolymers to large scale morphology, and their active mechanics is both driven by chiral components and serves to generate chiral morphologies. Increasingly such active chiral phenomena are being viewed through the lens of odd mechanics in both biological systems and in bio-inspired soft robotics. As many biological materials are localised to thin deformable interfaces such as membranes and filaments we consider the mechanics of active fluid membranes and active elastic rods, with a focus on odd contributions to the stress which break Maxwell-Betti reciprocity.

In the case of fluid membranes these generate geometric `odd elastic' forces in response to mean curvature gradients but directed perpendicularly. As a result, they induce tangential membrane flows that circulate around maxima and minima of membrane curvature. For shapes with anisotropy in the shape operator these flows can lead to linear instabilities. We describe examples for spheroids, membranes tubes and helicoids, discussing the relevance and predictions for left-right symmetry breaking in morphogenesis.

In the case of elastic rods we examine odd contributions to the bending moment which lead to unstable waves when inertia is non-negligible. We discuss the dispersion relations, coarsening dynamics and how these theories relate to experiments on odd robotic filaments.   

Confinement-induced phase transitions of Living Chiral Crystal

Active crystal is a unique manifestation of active matter characterized by crystalline order. While recent studies have shown how the interplay between activity and elasticity can result in intriguing self-organized dynamical states in active crystals, how these nonequilibrium phases and their excitations are influenced by boundary conditions remains unexplored. Living Chiral Crystal (LCC), self-assembled by swimming starfish embryos, represents an active crystal distinguished by non-reciprocal (chiral) interactions. Here we explore the influence of boundary interactions on the structure and dynamics of LCC. We confine LCC within regular polygonal and circular boundaries, while controlling the density of LCC within the confinements. We observe various phases of LCC, ranging from localized fluctuations to robust rigid body rotations and chaotic mixing. These findings provide insights into how LCC serves as a model system for uncovering novel and diverse properties of non-reciprocal active matter.   

Spontaneous self-excited odd elastic waves in living crystals

When chiral active particles interact with each other non-reciprocally and form a crystal, the displacement gradient and the stress that were independent in passive materials become coupled. The resulting asymmetric off-diagonal elements in the elastic moduli tensor represent odd elasticity. The living crystal of embryos of starfish Patiria miniata has recently been shown to exhibit odd elasticity, with the possible indirect observation of odd elastic wave. However, simple models for odd elastic waves are overdamped because of viscous forces arising from the longitudinal interaction among the embryos. In this work, we show how to detect odd elastic waves by measuring appropriate current correlation functions. Analysis of the experimental data using the current correlation function approach demonstrates that the oscillatory behavior experimentally observed in the living crystal was self-circling motion of the embryos. Nevertheless, we show that noise can excite the wave despite the presence of damping, and discuss the conditions required for the observation of a stochastic odd elastic wave in this system.

This work was partially supported by the National Science Foundation under Grant no. DMR-2006446.   

Chiral matters

The properties of materials, including their structure, shape, rheological response and mechanical stability arise spontaneously from the interactions between elementary building blocks. In passive materials these are often reciprocal, energy conserving and subject to thermal fluctuations. I will discuss the physics of out-of-equilibrium solids and fluids built out of mechanically spinning components. These active materials which blur the line between solid and fluid, spontaneously exhibit phenomena from chiral surface waves, to perling instabilities, to dislocation motility and crystal-whorl states. These rich and lively dynamics enable inference of the rheological properties of these odd materials which include odd stress, odd viscosity and odd elasticity.   

Nonreciprocity in active microbial populations

Active microbial populations are characterized by numerous interacting components, including active and passive elements. For example, animal groups physically interact with their environment and actively avoid predators, yet it's their social engagements that amplify the complexity of these systems. Intriguingly, the emerging concept of nonreciprocity offers fresh angles to untangle these complexities. In this presentation, I will summarize our recent experimental endeavors to understand nonreciprocity and its biological relevance. We have observed experimentally that nonreciprocal interactions among dense microbial populations, which include active and passive members, lead to traveling waves, arrested phase separation, and chiral edge states. I will also touch upon the potential evolutionary implications of these findings in population biology. Finally, I will illustrate the ubiquity of these responses in diverse microbial groups, from bacterial biofilms to nematodes.   

Microscopic nonreciprocity drives emergent dynamic phase of living crystals

Active matter systems can exhibit non-equilibrium phases under broken symmetries. Non-reciprocal interactions arising from broken action-reaction symmetry are hypothesized to be prevalent in natural systems, which calls for experimental platforms capable of modulating these interactions. To address this, we leverage on the new discovery of living chiral crystals composed of starfish embryos, and investigate mixtures of starfish embryos at two distinct developmental stages. Our observations reveal that the binary mixture undergoes a dynamic phase marked by spontaneous collective chiral movement. Moreover, extended observations over several hours indicate a transition from this dynamic state to a static crystalline phase. To elucidate these phenomena, we introduce a coarse-grained theoretical framework that accounts for both the emergence of the dynamic phase through imbalanced hydrodynamic force interactions and the exceptional phase transition driven by variations in embryo number densities. Collectively, our research provides a robust experimental platform for exploring emergent phases and phase transitions in active matter systems.   

Spontaneous Demixing of Binary Colloidal Flocks

Population heterogeneity is ubiquitous among active living systems, but little is known about its role in determining their spatial organization and large-scale dynamics. Combining evidence from synthetic active fluids assembled from self-propelled colloidal particles along with theoretical predictions at the continuum scale, we demonstrate the spontaneous demixing of binary polar liquids within circular confinement. Our analysis reveals how both active speed heterogeneity and non-reciprocal repulsive interactions lead to self-sorting behavior. By establishing general principles for the self-organization of binary polar liquids, our findings highlight the specificity of multi-component active systems.   

Non-Reciprocal Landau-Ginzburg Theories

The study of critical phases for systems out of equilibrium has a wide range of applicability including soft matter physics, open quantum system and biological systems. In critical phenomena at equilibrium, the study of the emergent conformal symmetry at a phase transition provides a way to systematically classify critical behavior. Unlike their equilibrium counterparts, non-equilibrium criticality does not in general lead to the standard conformal group as an emergent symmetry group. In particular, time and space do not scale in the same way which leads to a dynamical critical exponent, z, which differs from 1. This difference makes a systematic understanding of non-equilibrium criticality challenging. We show that non-reciprocal interactions can lead to an emergent sound mode that dominates the low energy behavior of the physical system. This emergent sound mode gives us a dynamical critical exponent z=1 and simplifies the study of critical phases. We use this to build non-reciprocal Landau-Ginzburg theories and discuss their physical properties.   

Statistical Mechanics of the Non-reciprocal Ising model near Zero Temperature

We study the kinetic Ising model with non-reciprocal interactions between spins, the maximum entropy distribution of binary variables conditioned on a past state with constraints on means and lagged correlations. As a maximum entropy model, the non-reciprocal Ising model offers a coarse-grained representation of biological systems, in particular the inherently non-reciprocal interactions of a neural system. We consider two non-reciprocally coupled 1D Ising chains in the low temperature regime. In this limit, the dynamics becomes deterministic, corresponding to a cellular automata update rule. Adjusting the coupling strength can alter the update rule, causing a dynamical phase transition in the absence of noise analogous to a quantum critical point of a thermodynamic system. Notably, as non-reciprocity increases through the transition at zero temperature, the system becomes spatially decoupled, inducing disorder at zero temperature. Close to the phase transition, we demonstrate the low temperature effective theory of the system is given by the driven Sine-Gordon equation. Previous studies have shown that the introduction of non-reciprocity can lead to the emergence of relevant sound modes in the long wavelength dynamics of certain systems, altering critical phenomena. A similar effect is evident in our model, where the dynamical critical exponent 'z' deviates from the expected value of 'z=2' in the reciprocal kinetic Ising model.   

Emergence of hidden nonreciprocity as a stabilizing effective potential in active matter

Nonreciprocal interactions are ubiquitous, ranging from colloidal particles to neurons to social dynamics. These nonconservative forces are known to induce complex oscillations between different states. Here, we identify a special class of systems evolving on a potential and under nonreciprocal interactions. These models are distinguished by their stationary and exactly solvable steady-state distributions of positions. Under thermal noise, an equilibrium Boltzmann distribution characterizes the steady state, showing no sign of nonreciprocity. Under exponentially correlated noise, which is characteristic of active matter, the nonreciprocity emerges in the steady-state distribution as an effective potential that stabilizes minima of the true potential. Thus, we reveal an unexpected interplay involving active fluctuations and reciprocal and nonreciprocal couplings. Our findings have interesting implications for the role and detection of nonreciprocity.   

The Three-Body Problem: Non-reciprocity and multibody interaction in acoustically levitated granular systems

When particles interact via a surrounding fluid medium, the coarse-grained interactions often break reciprocity. While recent advancements predominantly concentrate on effectively pairwise additive non-reciprocal interactions, non-reciprocity in fluid-mediated interactions inherently possesses a multibody character that cannot be linearly decomposed. Here, we highlight the inherent multibody and non-reciprocal nature of acoustic interactions, even in systems comprising as few as three particles. As in the classic gravitational three-body problem, a third particle greatly complicates the dynamics of the system; however, here it also renders non-reciprocal and non-pairwise effects dominant. In particular, we demonstrate that the interaction between two particles depends sensitively on the relative position of the third particle. We further show that this three-particle system harnesses energy from its non-reciprocal interaction, exhibiting emergent behavior such as self-propulsion and limit cycle. This minimal three-particle system not only explains how multibody, non-reciprocal interactions can lead to self-excitation in acoustic levitated systems but also offers a potential model for similar phenomena in other fluid-mediated interactions.   

Time-modulated Chiral Solids

The shape of passive solids and liquids is typically determined by the time-independent interactions between thermally fluctuating constituents. 

By contrast, in active and driven materials, the interactions can be modulated in time opening a new design axis. 

Using an external field, we drive magnetic colloids to rotate with a rotational frequency that is modulated in time, generating non-trivial structure. 

We show how the structures we observe are created as a result of time modulated bulk stresses and orientationally dependent surface forces.   

Dynamics of topological defects in the presence of non-reciprocal interactions: twisting, stabilisation and enhanced annihilation

We introduce an extension of the celebrated 2D XY model with non-reciprocal couplings, where short-range spin-spin interactions are weighted by a smooth anisotropic kernel. We investigate the dynamics of localised, non-reciprocal topological defects (vortices), responsible for the emergence of a Kosterlitz-Thouless transition in the reciprocal, equilibrium, case. We show that in addition to the topological charge, the actual shape of the defects, that can be characterized with a scalar, is crucial to faithfully describe their dynamics. Non-reciprocal interactions render some defect shapes unstable, twisting the field around them and radically affecting defect-defect interactions. Non-reciprocity leads to effective transverse forces, which at the defect/defect level, result in a variety of scenarios, such as very fast attraction and annihiliation, long-term stabilisation and spontaneous defect generation, strongly impacting the large-scale behaviour of the system.