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Two hallmarks of non-equilibrium systems, from active colloids to animal herds, are agents motility and nonreciprocal interactions. Their interplay creates feedback loops that lead to complex spatiotemporal dynamics crucial to understand and control the non-linear response of active systems. Here, we introduce a minimal model that captures these two features at the microscopic scale, while admitting an exact hydrodynamic theory valid also in the fully-nonlinear regime. We applied statistical mechanics techniques to exactly coarse-grain the microscopic model and used dynamical systems insights to analyze the resulting equations. In the absence of motility, we find two transitions to oscillatory phases occurring via distinct mechanisms: a Hopf bifurcation and a Saddle-Node on Invariant Circle (SNIC) bifurcation. In the presence of motility, this rigorous approach, complemented by numerical simulations, allows us to quantitatively assess the hitherto neglected impact of inter-species nonreciprocity on a paradigmatic transition in active matter: the emergence of collective motion. When nonreciprocity is weak, we show that flocking is accelerated and bands tend to synchronize with a spatial overlap controlled by nonlinearities. When nonreciprocity is strong, flocking is superseded by a Chase & Rest dynamical phase where each species alternates between a chasing state, when they propagate, and a resting state, when they stand still. Phenomenological models with linear non-reciprocal couplings fail to predict this Chase & Rest phase which illustrates the usefulness of our exact coarse-graining procedure.

Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spiking requires a fine-tuned ratio between local amplification and suppression strengths. But, in Nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking if the reacting components can spread out: even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking is a robust phenomenon that can occur even if the ratio between amplification and suppression does not allow spiking when the two sides are homogeneously mixed. We analytically derive a spiking phase diagram that depends on two parameters: (i) the ratio between the system size and the characteristic diffusive length-scale, and (ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator-prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction-diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media., Comment: 31 pages. 15 figures. Supplemental videos available at: https://home.uchicago.edu/~vitelli/videos.html

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales where the cascade is eventually arrested by dissipation. In this Letter, we show how to harness these seemingly structureless turbulent cascades to generate patterns. Conceptually, pattern or structure formation entails a process of wavelength selection: patterns typically arise from the linear instability of a homogeneous state. By contrast, the mechanism we propose here is fully non-linear and triggered by a non-dissipative arrest of turbulent cascades. Instead of being dissipated, energy piles up at intermediate scales. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns is set by a non-dissipative transport coefficient called odd or gyro viscosity. This non-dissipative viscosity is ubiquitous in chiral systems ranging from plasmas, bio-active media and quantum fluids. Beyond chiral fluids, cascade-induced pattern formation could occur in natural systems including oceanic and atmospheric flows, planetary and stellar plasma such as the solar wind, as well as in industrial processes such as the pulverization of objects into debris or in chemistry and biology where the coagulation of droplets can be interrupted at a tunable scale., Comment: 12 pages, 6 figures

Control of particle motion is generally achieved by applying an external field that acts directly on each particle. Here, we propose a global way to manipulate the motion of a particle by dynamically changing the properties of the fluid in which it is immersed. We exemplify this principle by considering a small particle sinking in an anisotropic fluid whose viscosity depends on the shear axis. In the Stokes regime, the motion of an immersed object is fully determined by the viscosity of the fluid through the mobility matrix, which we explicitly compute for a pushpin-shaped particle. Rather than falling upright under the force of gravity, as in an isotropic fluid, the pushpin tilts to the side, sedimenting at an angle determined by the viscosity anisotropy axis. By changing this axis, we demonstate control over the pushpin orientation as it sinks, even in the presence of noise, using a closed feedback loop. This strategy to control particle motion, that we dub viscous tweezers, could be experimentally realized in a fluid comprised of elongated molecules by suitably changing their global orientation., Comment: 5 pages, 4 figures + SI

Cellular form and function emerge from complex mechanochemical systems within the cytoplasm. No systematic strategy currently exists to infer large-scale physical properties of a cell from its many molecular components. This is a significant obstacle to understanding biophysical processes such as cell adhesion and migration. Here, we develop a data-driven biophysical modeling approach to learn the mechanical behavior of adherent cells. We first train neural networks to predict forces generated by adherent cells from images of cytoskeletal proteins. Strikingly, experimental images of a single focal adhesion protein, such as zyxin, are sufficient to predict forces and generalize to unseen biological regimes. This protein field alone contains enough information to yield accurate predictions even if forces themselves are generated by many interacting proteins. We next develop two approaches - one explicitly constrained by physics, the other more agnostic - that help construct data-driven continuum models of cellular forces using this single focal adhesion field. Both strategies consistently reveal that cellular forces are encoded by two different length scales in adhesion protein distributions. Beyond adherent cell mechanics, our work serves as a case study for how to integrate neural networks in the construction of predictive phenomenological models in cell biology, even when little knowledge of the underlying microscopic mechanisms exist., Comment: 30 pages, 7 figures

Odd elasticity encompasses active elastic systems whose stress-strain relationship is not compatible with a potential energy. As the requirement of energy conservation is lifted from linear elasticity, new anti-symmetric (odd) components appear in the elastic tensor. In this work, we study the odd elasticity and non-Hermitian wave dynamics of active surfaces, specifically plates of moderate thickness. We find that a free-standing moderately thick, isotropic plate can exhibit two odd-elastic moduli, both of which are related to shear deformations of the plate. These odd moduli can endow the vibrational modes of the plate with a nonzero topological invariant known as the first Chern number. Within continuum elastic theory, we show that the Chern number is related to the presence of unidirectional shearing waves that are hosted at the plate's boundary. We show that the existence of these chiral edge waves hinges on a distinctive two-step mechanism: the finite thickness of the sample gaps the shear modes and the odd elasticity endows them with chirality., 13 pages, 6 figures

We study a class of bifurcations generically occurring in dynamical systems with non-mutual couplings ranging from minimal models of coupled neurons to predator-prey systems and non-linear oscillators. In these bifurcations, extended attractors such as limit cycles, limit tori and strange attractors merge and split in a similar way as fixed points in a pitchfork bifurcation. We show that this merging and splitting coincides with the coalescence of covariant Lyapunov vectors, generalizing the notion of exceptional points to non-linear dynamical systems. We distinguish two classes of systems, corresponding respectively to continuous and discontinuous behaviors of the covariant Lyapunov vectors at the transition. We illustrate our results with concrete examples from neuroscience, ecology, and physics. Potential applications of our work include the perceptual bistability human subjects experience when presented with certain temporally periodic stimuli, non-linear oscillations of the relative abundance of competing species in ecosystems, as well as PT-symmetry-breaking phase transitions in coupled lasers and driven-dissipative quantum condensates., 15 pages, 10 figures

We provide a perspective on soft metamaterials and biomatter with a special emphasis on future research directions at the interface between material intelligence and adaptation., 7 pages, 2 figures

The flow of momentum and energy in a fluid is typically associated with dissipative transport coefficients: viscosity and thermal conductivity. Fluids that break certain symmetries such as mirror symmetry and time-reversal invariance can display non-dissipative transport coefficients called odd (or Hall) viscosities and thermal conductivities. The goal of this paper is to elucidate the microscopic origin of these dissipationless transport coefficients using kinetic theory. We show that odd viscosity and odd thermal conductivity arise when the linearized collision operator is not Hermitian. This symmetry breaking occurs when collisions are chiral, i.e. not mirror symmetric. To capture this feature in a minimalistic way, we introduce a modified relaxation time approximation in which the distribution function is rotated by an angle characterizing the average chirality of the collisions. In the limit of an infinitesimal rotation, the effect of the parity-violating collisions can be described as an emergent effective magnetic field., 5 pages, 3 figures + SI

Elasticity typically refers to a material's ability to store energy, while viscosity refers to a material's tendency to dissipate it. In this review, we discuss fluids and solids for which this is not the case. These materials display additional linear response coefficients known as odd viscosity and odd elasticity. We first introduce odd elasticity and odd viscosity from a continuum perspective, with an emphasis on their rich phenomenology, including transverse responses, modified dislocation dynamics, and topological waves. We then provide an overview of systems that display odd viscosity and odd elasticity. These systems range from quantum fluids, to astrophysical gasses, to active and driven matter. Finally, we comment on microscopic mechanisms by which odd elasticity and odd viscosity arise., Comment: 30 pages, 7 figures

Active matter composed of energy-generating microscopic constituents is a promising platform to create autonomous functional materials. However, the very presence of these microscopic energy sources is what makes active matter prone to dynamical instabilities and hence hard to control. Here, we show that these instabilities can be coaxed into work-generating limit cycles that turn active matter into robots. We illustrate this general principle in odd active media, model systems whose interaction forces are as simple as textbook molecular bonds yet not constrained to be the gradient of a potential. These emergent robotic functionalities are demonstrated by revisiting what is arguably the oldest of inventions: the wheel. Unlike common wheels that are driven by external torques, an odd wheel undergoes work-generating limit cycles that allow it to roll autonomously uphill by virtue of its own deformation, as demonstrated by our prototypes. Similarly, familiar scattering phenomena, like a ball bouncing off a wall, turn into basic robotic manipulations when either the ball or the wall is odd. Using continuum mechanics, we reveal collective robotic mechanisms that steer the outcome of collisions or influence the absorption of impacts in experiments. Beyond robotics, work-generating limit cycles can also control the non-linear dynamics of active soft materials, biological systems and driven nanomechanical devices., 4 Figures, Methods and SI. Supplemental videos at https://home.uchicago.edu/~vitelli/videos.html

Statistical mechanics provides the foundation for describing complex materials using only a few thermodynamic variables. No such framework currently exists far from equilibrium. In this Letter, we demonstrate how thermodynamics emerges far from equilibrium, using fluids composed of active spinners as a case study. Activity gives rise to a single effective temperature that parameterizes both the equation of state and the emergent Boltzmann statistics. The same effective temperature, renormalized by velocity correlations, controls the linear response through canonical Green-Kubo relations for both the familiar shear viscosity and the odd (or Hall) viscosity observed in chiral fluids. The full frequency dependence of these viscosities can be derived analytically by modeling the active-spinner fluid as a random walker undergoing cyclotron motion in shear-stress space. More generally, we provide a first-principles derivation of the Green-Kubo relations valid for a broader class of fluids far from equilibrium. Besides advancing non-equilibrium thermodynamics, our work demonstrates in silico a non-invasive microrheology of active fluids., Supplementary movies at https://www.youtube.com/playlist?list=PLnzJw7B7T8cKLck3wSoMRJVTSFSdrExeh

Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials., 15 pages including Supplementary Information, 5 figures. See https://www.youtube.com/playlist?list=PLnzJw7B7T8cIc1EjLVq3AU_hOe1PEAdt9 for Supplementary Movies

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden non-spatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such non-spatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of non-spatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena., Comment: 5 pages, 3 figures + SI

Active materials are capable of converting free energy into mechanical work to produce autonomous motion, and exhibit striking collective dynamics that biology relies on for essential functions. Controlling those dynamics and transport in synthetic systems has been particularly challenging. Here, we introduce the concept of spatially structured activity as a means to control and manipulate transport in active nematic liquid crystals consisting of actin filaments and light-sensitive myosin motors. Simulations and experiments are used to demonstrate that topological defects can be generated at will, and then constrained to move along specified trajectories, by inducing local stresses in an otherwise passive material. These results provide a foundation for design of autonomous and reconfigurable microfluidic systems where transport is controlled by modulating activity with light.

Folding mechanisms are zero elastic energy motions essential to the deployment of origami, linkages, reconfigurable metamaterials and robotic structures. In this paper, we determine the fate of folding mechanisms when such structures are miniaturized so that thermal fluctuations cannot be neglected. First, we identify geometric and topological design strategies aimed at minimizing undesired thermal energy barriers that generically obstruct kinematic mechanisms at the microscale. Our findings are illustrated in the context of a quasi one-dimensional linkage structure that harbors a topologically protected mechanism. However, thermal fluctuations can also be exploited to deliberately lock a reconfigurable metamaterial into a fully expanded configuration, a process reminiscent of order by disorder transitions in magnetic systems. We demonstrate that this effect leads certain topological mechanical structures to exhibit an abrupt change in the pressure -- a bulk signature of the underlying topological invariant at finite temperature. We conclude with a discussion of anharmonic corrections and potential applications of our work to the the engineering of DNA origami devices and molecular robots., Comment: 15 pages, 8 figures

The properties of small clusters can differ dramatically from the bulk phases of the same constituents. In equilibrium, cluster assembly has been recently explored, whereas out of equilibrium, the physical principles of clustering remain elusive. These principles underlie phenomena from molecular assembly to the formation of planets from granular matter. Here, we introduce acoustic levitation as a platform to experimentally probe the formation of nonequilibrium small structures in a controlled environment. We focus on the minimal models of cluster formation: six and seven millimetre-scale particles in two dimensions. Experiments and modelling reveal that, in contrast to thermal colloids, in non-equilibrium granular ensembles the magnitude of active fluctuations controls not only the assembly rates but also their assembly pathways and ground-state statistics. These results open up new possibilities for non-invasively manipulating macroscopic particles, tuning their interactions, and directing their assembly.

Topological states can be used to control the mechanical properties of a material along an edge or around a localized defect. The surface rigidity of elastic networks is characterized by a bulk topological invariant called the polarization; materials with a well-defined uniform polarization display a dramatic range of edge softnesses depending on the orientation of the polarization relative to the terminating surface. However, in all three-dimensional mechanical metamaterials proposed to date, the topological edge modes are mixed with bulk soft modes and so-called Weyl loops. Here, we report the design of a gapped 3D topological metamaterial with a uniform polarization that displays a corresponding asymmetry between the number of soft modes on opposing surfaces and, in addition, no bulk soft modes. We then use this construction to localize topological soft modes in interior regions of the material by including defect structures---dislocation loops---that are unique to three dimensions. We derive a general formula that relates the difference in the number of soft modes and states of self-stress localized along the dislocation loop to the handedness of the vector triad formed by the lattice polarization, Burgers vector, and dislocation-line direction. Our findings suggest a novel strategy for pre-programming failure and softness localized along lines in 3D, while avoiding extended periodic failure modes associated with Weyl loops., Comment: 23 pages, 4 figures

We investigate how material rigidity acts as a key control parameter for the failure of solids under stress. In both experiments and simulations, we demonstrate that material failure can be continuously tuned by varying the underlying rigidity of the material while holding the amount of disorder constant. As the rigidity transition is approached, failure due to the application of uniaxial stress evolves from brittle cracking to system-spanning diffuse breaking. This evolution in failure behavior can be parameterized by the width of the crack. As a system becomes more and more floppy, this crack width increases until it saturates at the system size. Thus, the spatial extent of the failure zone can be used as a direct probe for material rigidity., 12 pages, 4 figures

We present a theoretical study of the energetics of thin nematic shells with two charge one-half defects and one charge-one defect. We determine the optimal arrangement: the defects are located on a great circle at the vertices of an isosceles triangle with angles of 66 degrees at the charge one-half defects and a distinct angle of 48 degrees, consistent with experimental findings. We also analyse thermal fluctuations around this ground state and estimate the energy as a function of thickness. We find that the energy of the three-defect shell is close to the energy of other known configurations having two charge-one and four charge one-half defects. This finding, together with the large energy barriers separating one configuration from the others, explains their observation in experiments as well as their long-time stability., Comment: 8 pages, 7 figures

This review introduces the elasticity theory of two-dimensional crystals and nematic liquid crystals on curved surfaces, the energetics of topological defects (disclinations, dislocations and pleats) in these ordered phases, and the interaction of defects with the underlying curvature. This chapter concludes with two cases of three-dimensional nematic phases confined to spaces with curved boundaries, namely a torus and a spherical shell., Comment: Book chapter. 30 pages, 11 figures

A topological mechanism is a zero elastic-energy deformation of a mechanical structure that is robust against smooth changes in system parameters. Here, we map the nonlinear elasticity of a paradigmatic class of topological mechanisms onto linear fermionic models using a supersymmetric field theory introduced by Witten and Olive. Heuristically, this approach consists of taking the square root of a non-linear Hamiltonian and generalizes the standard procedure of obtaining two copies of Dirac equation from the square root of the linear Klein Gordon equation. Our real space formalism goes beyond topological band theory by incorporating non-linearities and spatial inhomogeneities, such as domain walls, where topological states are typically localized. By viewing the two components of the real fermionic field as site and bond displacements respectively, we determine the relation between the supersymmetry transformations and the Bogomolny-Prasad-Sommerfield (BPS) bound saturated by the mechanism. We show that the mechanical constraint, which enforces a BPS saturated kink into the system, simultaneously precludes an anti-kink. This mechanism breaks the usual kink-antikink symmetry and can be viewed as a manifestation of the underlying supersymmetry being half-broken., Comment: 14 pages, 5 figures

We compare the harmonic and anharmonic properties of the vibrational modes in 3-dimensional jammed packings of frictionless spheres interacting via repulsive, finite range potentials. A crossover frequency is apparent in the density of states, the diffusivity and the participation ratio of the modes. At this frequency, which shifts to zero at the jamming threshold, the vibrational modes have a very small participation ratio implying that the modes are quasi-localized. The most anharmonic modes occur at low frequency which is opposite to what is normally found in crystals. The lowest frequency modes have the strongest response to the pressure and the lowest energy barriers to mechanical failure., Comment: 6 pages, 5 figures

We present a systematic study of how vortices in superfluid films interact with the spatially varying Gaussian curvature of the underlying substrate. The Gaussian curvature acts as a source for a geometric potential that attracts (repels) vortices towards regions of negative (positive) Gaussian curvature independently of the sign of their topological charge. Various experimental tests involving rotating superfluid films and vortex pinning are first discussed for films coating gently curved substrates that can be treated in perturbation theory from flatness. An estimate of the experimental regimes of interest is obtained by comparing the strength of the geometrical forces to the vortex pinning induced by the varying thickness of the film which is in turn caused by capillary effects and gravity. We then present a non-perturbative technique based on conformal mappings that leads an exact solution for the geometric potential as well as the geometric correction to the interaction between vortices. The conformal mapping approach is illustrated by means of explicit calculations of the geometric effects encountered in the study of some strongly curved surfaces and by deriving universal bounds on their strength., Comment: 50 pages, 38 figures