The theory of wave turbulence describes the properties of physical systems composed of a set of weak-amplitude random waves interacting nonlinearly. Here, we study odd wave turbulence, which arises in chiral media subjected to non-reciprocal stresses, notably odd viscosity and odd elasticity. In both cases, we consider simple models for which we can derive and solve analytically the kinetic equations describing the long-term statistical behavior of spectral quantities such as energy or wave action. For odd viscosity, we consider a three-dimensional model that exhibits wave turbulence involving three-wave interactions, which gives rise to a direct energy cascade characterized by an anisotropic Kolmogorov-Zakharov (KZ) spectrum. For odd elasticity, we consider a quasi-one-dimensional overdamped model that exhibits much slower dynamics involving six-wave interactions. In that case, the KZ spectrum corresponding to a forward cascade of a conserved quantity we call odd energy, is nonlocal and therefore does not constitute a physical solution. However, the other KZ solution, which describes an inverse cascade of wave action, is only marginally non-local and is therefore valid up to a logarithmic correction. These two analytical theories provide a rigorous interpretation of direct numerical simulations, where the KZ spectrum is observed both in the case of odd viscosity (forward cascade) and of odd elasticity (inverse cascade).

Various biological and synthetic media out of equilibrium can be viewed as many-ratchet systems that rectify environmental noise through local measurements and information processing, like Maxwell's prototypical demon. These systems pose a challenge to standard coarse-graining approaches because they are described in terms of decision-making protocols similar to computer programs rather than force laws. Here, we study a many-body generalization of the Maxwell demon problem: a fluid composed of adaptive particles that achieve collective behavior by biasing noise-driven scattering events subject to measurements. Using a combination of information-theoretic, kinetic, and hydrodynamic tools, we elucidate how microscopic decision-making protocols, rather than microscopic forces, generate macroscopic active states sustained by continuous measurements. These include an informational version of flocking whose order parameter is bounded by the information measured, and the onset of which may be viewed as a measurement-induced phase transition. We find that the signature of such microscopic choices is an "informational activity" that selectively compresses phase space, without work, and causes deviations from equilibrium scaling with the magnitude of environmental noise. We envision applications to noise-induced patterning performed by collections of microrobots guided by reinforcement learning or programmable phoretic colloids in turbulent flows that exploit local measurements and control actions to counteract the scrambling of information by chaos.

Reciprocity is a fundamental symmetry present in many natural phenomena and engineered systems. Distinct situations where this symmetry is broken are typically grouped under the umbrella term "nonreciprocity", colloquially defined by: the action of A on B $\neq$ the action of B on A. In this review, we elucidate what nonreciprocity is by providing an introduction to its most salient classes: nonvariational dynamics, violations of Newton's third law, broken detailed balance, nonreciprocal responses and nonreciprocity of arbitrary linear operators. Next, we point out where to find these manifestations of non-reciprocity, from ensembles of particles with field mediated interactions to synthetic neural networks and open quantum systems. Given this breadth of contexts and the lack of an all-encompassing definition, it makes it all the more intriguing that some general conclusions can be gathered, when distinct definitions of nonreciprocity overlap. We explore what these universal consequences are with a special emphasis on collective phenomena that arise in nonreciprocal many-body systems. The topics covered include nonreciprocal phase transitions and non-normal amplification of noise and perturbations. We conclude with some open questions.

Computation is the controlled evolution of a state. Asynchronous evolutions, where all parts of the state change in their own time without stopping each other, put this control in jeopardy. It is in fact a mystery how natural processes perform asynchronous computations using many units with no global orchestration. Here we demonstrate how collective computational abilities can emerge in asynchronous many-body systems. The key insight is to split the physical "hardware" underlying the computation into two asymmetrically coupled parts, analogous to position and momentum in a harmonic oscillator. The resulting inertia nudges the evolution of the state so that the asynchronous computation proceeds in the right order. By treating our inertial asynchronous computer as a nonequilibrium material, we map out its phase diagram numerically and analytically using a framework we dub loop dynamical mean-field theory. We experimentally demonstrate our approach using analog spiking neuromorphic chips designed to mimic actual neurons in the brain. In addition, we construct software that can run on asynchronous hardware: we denoise movies whose clean versions were never seen during training, an instantiation of the generalization transition underlying modern machine learning. Our results point to a general strategy for reliable, decentralized computation in energy-constrained settings from dynamics self-assembly to cell differentiation.

Living systems display complex behaviors driven by physical forces as well as decision-making. Hydrodynamic theories hold promise for simplified universal descriptions of socially-generated collective behaviors. However, the construction of such theories is often divorced from the data they should describe. Here, we develop and apply a data-driven pipeline that links micromotives to macrobehavior by augmenting hydrodynamics with individual preferences that guide motion. We illustrate this pipeline on a case study of residential dynamics in the United States, for which census and sociological data is available. Guided by Census data, sociological surveys, and neural network analysis, we systematically assess standard hydrodynamic assumptions to construct a sociohydrodynamic model. Solving our simple hydrodynamic model, calibrated using statistical inference, qualitatively captures key features of residential dynamics at the level of individual US counties. We highlight that a social memory, akin to hysteresis in magnets, emerges in the segregation-integration transition even with memory-less agents. This suggests an explanation for the phenomenon of neighborhood tipping, whereby a small change in a neighborhood's population leads to a rapid demographic shift. Beyond residential segregation, our work paves the way for systematic investigations of decision-guided motility in real space, from micro-organisms to humans, as well as fitness-mediated motion in more abstract genomic spaces.

Nonreciprocal interactions in many-body systems lead to time-dependent states, commonly observed in biological, chemical, and ecological systems. The stability of these states in the thermodynamic limit and the critical behavior of the phase transition from static to time-dependent states are not yet fully understood. To address these questions, we study a minimalistic system endowed with nonreciprocal interactions: an Ising model with two spin species having opposing goals. The mean-field equation predicts three stable phases: disorder, static order, and a time-dependent swap phase. Large-scale numerical simulations support the following: (i) in two dimensions, the swap phase is destabilized by defects; (ii) in three dimensions, the swap phase is stable and has the properties of a time crystal; (iii) the transition from disorder to swap in three dimensions is characterized by the critical exponents of the 3D XY model and corresponds to the breaking of a continuous symmetry, time translation invariance; (iv) when the two species have fully antisymmetric couplings, the static-order phase is unstable in any finite dimension due to droplet growth; and (v) in the general case of asymmetric couplings, static order can be restored by a droplet-capture mechanism preventing the droplets from growing indefinitely. We provide details on the full phase diagram, which includes first- and second-order-like phase transitions, and study how the system coarsens into swap and static-order states.

Tissue dynamics play a crucial role in biological processes ranging from wound healing to morphogenesis. However, these noisy multicellular dynamics are notoriously hard to predict. Here, we introduce a biomimetic machine learning framework capable of inferring noisy multicellular dynamics directly from experimental movies. This generative model combines graph neural networks, normalizing flows and WaveNet algorithms to represent tissues as neural stochastic differential equations where cells are edges of an evolving graph. This machine learning architecture reflects the architecture of the underlying biological tissues, substantially reducing the amount of data needed to train it compared to convolutional or fully-connected neural networks. Taking epithelial tissue experiments as a case study, we show that our model not only captures stochastic cell motion but also predicts the evolution of cell states in their division cycle. Finally, we demonstrate that our method can accurately generate the experimental dynamics of developmental systems, such as the fly wing, and cell signaling processes mediated by stochastic ERK waves, paving the way for its use as a digital twin in bioengineering and clinical contexts.

Disordered systems generically exhibit aging and a glass transition. Previous studies have long suggested that nonreciprocity tends to destroy glassiness. Here, we show that this is not always the case using a bipartite spherical Sherrington-Kirkpatrick model that describes the antagonistic coupling between two identical complex agents modeled as macroscopic spin glasses. Our dynamical mean-field theory calculations reveal an exceptional-point-mediated transition from a static disorder phase to an oscillating amorphous phase as well as nonreciprocal aging with slow dynamics and oscillations.

Self-assembled systems, from synthetic nanostructures to developing organisms, are composed of fluctuating units capable of forming robust functional structures despite noise. In this Letter we ask: are there fundamental bounds on the robustness of self-organized nano-systems? By viewing self-organization as noisy encoding, we prove that the positional information capacity of short-range classical systems with discrete states obeys a bound reminiscent of area laws for quantum information. This universal bound can be saturated by fine-tuning transport coefficients. When long-range correlations are present, global constraints reduce the need for fine-tuning by providing effective integral feedback. Our work identifies bio-mimetic principles for the self-assembly of synthetic nanosystems and rationalizes, on purely information-theoretic grounds, why scale separation and hierarchical structures are common motifs in biology.

When the time-reversal and parity symmetries in a fluid are broken, transverse transport coefficients can arise in response to perturbations, an example being odd viscosity. We refer to these systems as odd fluids. While much progress has been made in the continuum theory of odd-viscous fluids, and non-collisional models for odd viscous fluids have been proposed, a classical microscopic description in which the transverse responses originate from collisions is lacking. In this paper, we show that a dilute granular gas of rough and inelastic particles driven by a constant torque is a minimal microscopic model of an odd fluid. By applying the methods of Boltzmann kinetic theory, we obtain a hydrodynamic description of the microscopic model. Then, using the method of adiabatic elimination, we numerically compute all the response coefficients of the model, explicitly showing that the model has many odd response terms. Our theory predicts that certain odd response coefficients can change sign even when the direction of the external torque is fixed. While we choose a particular case, the procedure we present can be applied to any collisional model. We also present a semi-quantitative method to determine the hydrodynamic variables of the theory by observing the eigenvalue spectrum of the linear collision operator.

Wave turbulence describes the statistical dynamics of weakly interacting waves in out-of-equilibrium systems. These systems are out of equilibrium because forcing and dissipation occur at different scales, leading to a flux across scales known as a turbulent cascade. However, the underlying medium is usually assumed to be passive. Here, we study situations where a turbulent cascade takes place on top of a driven-dissipative steady-state, which is already out of equilibrium without forcing. We focus on chiral active media, in which waves arise as a consequence of non-reciprocal responses known as odd viscosity and odd elasticity. Combining analytical theory with large-scale direct numerical simulations, we show that fluids with odd viscosity sustain weak wave turbulence down to the smallest active scales, leading to anisotropic Kolmogorov-Zakharov spectra and a modified forward cascade. In the odd elastic solids, two new conserved quantities emerge (which we call wave action and odd energy), giving rise to an inverse cascade in the weak regime and a forward cascade in the strong regime. Our results pave the way towards understanding the chaotic nonlinear regime of systems with nonreciprocal responses from chiral plasma to engineered solids.

We study a class of bifurcations generically occurring in dynamical systems with nonmutual couplings ranging from models of coupled neurons to predator-prey systems and nonlinear 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 coincide with the coalescence of covariant Lyapunov vectors with vanishing Lyapunov exponents, a feature that generalizes the exceptional points that can exist in families of non-Hermitian matrices or operators. We distinguish two classes of bifurcations associated with generalized exceptional points, corresponding respectively to continuous and discontinuous behaviors of the covariant Lyapunov vectors at the transition depending on the presence of a Z_{2} symmetry. We outline some physical consequences of this class of theories exhibiting generalized exceptional points, including nonreciprocal responses, the destruction of isochrons, and anomalous noise effects. In particular, we show that the effective diffusion coefficient on the attractor can stay finite or even diverge when the noise strength vanishes. We illustrate our results with concrete examples from neuroscience, ecology, and physics.

Two hallmarks of non-equilibrium systems, from active colloids to animal herds, are agents motility and nonreciprocal interactions. Their interplay creates feedback loops leading to complex spatiotemporal dynamics crucial to understand and control the nonlinear 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. Our goal is to account for the fact that animal herds and colloidal swarms are rarely in the thermodynamic limit where particle number fluctuations can be completely ignored. Using statistical mechanics techniques we exactly coarse-grain a nonreciprocal microscopic model into a fluctuating hydrodynamics and use 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. Finally, we demonstrate how fluctuations in finite systems can be harnessed to characterize microscopic non-reciprocity from macroscopic time-correlation functions, even in phases where nonreciprocal interactions do not affect the thermodynamic steady-state.

Nonreciprocal interactions are widely observed in nonequilibrium systems, from biological or sociological dynamics to open quantum systems. Despite the ubiquity of nonreciprocity, its impact on phase transitions is not fully understood. In this work, we derive criteria to perturbatively assess whether nonreciprocity changes the universality class of pairs of asymmetrically coupled systems undergoing a phase transition. These simple criteria are stated in terms of the unperturbed critical exponents, in the spirit of the Harris criterion for disordered systems, and agree with numerical simulations. Beyond nonreciprocity, our approach provides guidelines for assessing how dynamical phase transitions are affected by perturbations.

Disordered systems can exhibit a dramatic slowdown of their dynamics called aging. Contrary to the established understanding that this phenomenon is destroyed by nonreciprocal interactions, we here show that the outcome crucially depends on the structure of the system. Unlike previous studies, which focused on random nonsymmetric interactions between simple microscopic components, we investigate a scenario where nonreciprocally coupled agents are macroscopic entities with complex internal dynamics, modeled as two identical spin glasses. This framework could be relevant for many biological systems, in which nonreciprocal interactions can arise at a coarse-grained level. Our dynamical mean-field theory calculations reveal a finite temperature transition from a static disordered phase to a non-time-translationally-invariant regime. Below this transition, mediated by a spectral singularity known as exceptional points, we find macroscopic oscillations superimposed on aging behavior. Asymptotically, the system rotates in the plane spanned by the two lowest energy modes of the uncoupled system. We contrast these results to the case of random nonreciprocity, where aging is suppressed at any finite temperature, and propose that the two cases correspond to two broader classes of systems, with "microscopic" versus "macroscopic" nonreciprocity, with aging surviving only in the second case.

Phonons are quasiparticles associated with mechanical vibrations in materials. They are at the root of the propagation of sound and elastic waves, as well as of thermal phenomena, which are pervasive in our everyday life and in many technologies. The fundamental understanding and control of phonon responses in natural and artificial media are key in the context of communications, isolation, energy harvesting and control, sensing and imaging. It has recently been realized that controlling different symmetry classes at the microscopic and mesoscopic scales in synthetic media offers a powerful tool to precisely tailor phononic responses for advanced acoustic and elastodynamic wave control. In this Review, we survey the recent progress in the design and synthesis of artificial phononic media, namely phononic crystals and metamaterials, guided by symmetry principles. Starting from tailored broken spatial symmetries, we discuss their interplay with time symmetries for non-reciprocal and non-conservative phenomena. We also address broader concepts that combine multiple symmetry classes to induce exotic phononic wave transport. We conclude with an outlook on future research directions based on symmetry engineering for the advanced control of phononic waves. Broken and tailored symmetries have a fundamental role in wave phenomena and their applications. This Review surveys the recent progress in the domain of artificial phononic media with an emphasis on the role of symmetry breaking, in both space and time, for advanced wave phenomena.

Systems with nonreciprocal interactions generically display time-dependent states. These are routinely observed in finite systems, from neuroscience to active matter, in which globally ordered oscillations exist. However, the stability of these uniform nonreciprocal phases in noisy spatially-extended systems, their fate in the thermodynamic limit, and the critical behavior of the corresponding phase transitions are not fully understood. Here, we address these questions by introducing a nonreciprocal generalization of the Ising model and study its phase transitions by means of numerical and analytical approaches. While the mean-field equations predict three stable homogeneous phases (disordered, ordered and a time-dependent swap phase), our large scale numerical simulations reveal a more complex picture. Static order is destroyed in any finite dimension due to the growth of rare droplets unless the symmetry between the two spin types is broken triggering a stabilizing droplet-capture mechanism. The swap phase is destroyed by fluctuations in two dimensions through the proliferation of spiral defects but stabilized in three dimensions where nonreciprocity changes the critical exponents from Ising to XY, thus giving rise to a robust spatially-distributed clock.

Chiral processes that lack mirror symmetry pervade nature from enantioselective molecular interactions to the asymmetric development of organisms. An outstanding challenge at the interface between physics and biology consists in bridging the multiple scales between microscopic and macroscopic chirality. Here, we combine theory, experiments and modern inference algorithms to study a paradigmatic example of dynamic chirality transfer across scales: the generation of tissue-scale flows from subcellular forces. The distinctive properties of our microscopic graph model and the corresponding coarse-grained viscoelasticity are that (i) net cell proliferation is spatially inhomogeneous and (ii) cellular dynamics cannot be expressed as an energy gradient. To overcome the general challenge of inferring microscopic model parameters from noisy high-dimensional data, we develop a nudged automatic differentiation algorithm (NADA) that can handle large fluctuations in cell positions observed in single tissue snapshots. This data-calibrated microscopic model quantitatively captures proliferation-driven tissue flows observed at large scales in our experiments on fibroblastoma cell cultures. Beyond chirality, our inference algorithm can be used to extract interpretable graph models from limited amounts of noisy data of living and inanimate cellular systems such as networks of convection cells and flowing foams.

The dynamics of many-body systems can often be captured in terms of only a few relevant variables. Mathematical and numerical approaches exist to identify these variables by exploiting a separation of time scales between slow relevant and fast irrelevant variables, but such a separation of scales is not always obvious or even available. In this work, we introduce an information-theoretic framework for dimensionality reduction in dynamical systems that bypasses this limitation by instead identifying relevant variables based on how predictive they are of the system's future. To do so, we mathematically formalize the intuition that model reduction is about keeping "relevant" information while throwing away "irrelevant" information. We characterize the solution of the resulting optimization problem and prove that it reduces to standard approaches when a separation of time scales is indeed present in the dynamics. Importantly, we find that within this framework, the problems of identifying relevant variables and identifying their effective dynamics decouple and may be solved separately. This makes the method tractable in practice and enables us to derive dimensionally-reduced variables from data with neural networks. Combined with existing equation learning methods, the procedure introduced in this work reveals the dynamical rules governing the system's evolution in a data-driven manner. We illustrate these tools in diverse settings including simulated chaotic systems, uncurated satellite recordings of atmospheric fluid flows, and experimental videos of cyanobacteria colonies in which we discover an emergent synchronization order parameter.

Physical learning is an emerging paradigm in science and engineering whereby (meta)materials acquire desired macroscopic behaviors by exposure to examples. So far, it has been applied to static properties such as elastic moduli and self-assembled structures encoded in minima of an energy landscape. Here, we extend this paradigm to dynamic functionalities, such as motion and shape change, that are instead encoded in limit cycles or pathways of a dynamical system. We identify the two ingredients needed to learn time-dependent behaviors irrespective of experimental platforms: (i) learning rules with time delays and (ii) exposure to examples that break time-reversal symmetry during training. After providing a hands-on demonstration of these requirements using programmable LEGO toys, we turn to realistic particle-based simulations where the training rules are not programmed on a computer. Instead, we elucidate how they emerge from physico-chemical processes involving the causal propagation of fields, like in recent experiments on moving oil droplets with chemotactic signalling. Our trainable particles can self-assemble into structures that move or change shape on demand, either by retrieving the dynamic behavior previously seen during training, or by learning on the fly. This rich phenomenology is captured by a modified Hopfield spin model amenable to analytical treatment. The principles illustrated here provide a step towards von Neumann's dream of engineering synthetic living systems that adapt to the environment.