Nonreciprocity as a generic route to traveling states

Nanocrystal Assemblies: Current Advances and Open Problems

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

Nonreciprocity and odd viscosity in chiral active fluids

Odd viscosity couples stress to strain rate in a dissipationless way. It has been studied in plasmas under magnetic fields, superfluid [Formula: see text], quantum-Hall fluids, and recently in the context of chiral active matter. In most of these studies, odd terms in the viscosity obey Onsager reciprocal relations. Although this is expected in equilibrium systems, it is not obvious that Onsager relations hold in active materials. By directly coarse-graining the kinetic energy and independently using both the Poisson-bracket formalism and a kinetic theory derivation, we find that the appearance of a nonvanishing angular momentum density, which is a hallmark of chiral active materials, necessarily breaks Onsager reciprocal relations. This leads to a non-Hermitian dynamical matrix for the total hydrodynamic momentum and to the appearance of odd viscosity and other nondissipative contributions to the viscosity. Furthermore, by accounting for both the angular momentum density and interactions that lead to odd viscosity, we find regions in the parameter space in which 3D odd mechanical waves propagate and regions in which they are mechanically unstable. The lines separating these regions are continuous lines of exceptional points, suggesting a possible nonreciprocal phase transition.

Nonequilibrium dynamics and entropy production of a trapped colloidal particle in a complex nonreciprocal medium

We discuss the two-dimensional motion of a Brownian particle that is confined to a harmonic trap and driven by a shear flow. The surrounding medium induces memory effects modeled by a linear, typically nonreciprocal coupling of the particle coordinates to an auxiliary (hidden) variable. The system's behavior resulting from the microscopic Langevin equations for the three variables is analyzed by means of exact moment equations derived from the Fokker-Planck representation, and numerical Brownian dynamics simulations. Increasing the shear rate beyond a critical value we observe, for suitable coupling scenarios with nonreciprocal elements, a transition from a stationary to a nonstationary state, corresponding to an escape from the trap. We analyze this behavior, analytically and numerically, in terms of the associated moments of the probability distribution, and from the perspective of nonequilibrium thermodynamics. Intriguingly, the entropy production rate remains finite when crossing the stability threshold.

Quantum Criticality in Open Quantum Spin Chains with Nonreciprocity

We investigate the impact of nonreciprocity on universality and critical phenomena in open quantum interacting many-body systems. Nonreciprocal open quantum systems often have an exotic spectral sensitivity to boundary conditions, known as the Liouvillian skin effect (LSE). By considering an open quantum XXZ spin chain that exhibits LSE, we demonstrate the existence of a universal scaling regime that is not affected by the presence of the LSE. We resolve the critical exponents, which differ from those of free fermions, via tensor network methods and demonstrate that observables exhibit a universal scaling collapse, irrespective of the reciprocity. We find that the LSE only becomes relevant when a healing length scale ξ_{heal} at the system's edge (which is different from the localization length of the eigenstate of the Liouvillian) exceeds the system size, allowing edge properties to dominate the physics. We expect this result to be a generic feature of nonreciprocal models in the vicinity of a critical point. The driven-dissipative quantum criticality we observe has no classical analog and stems from the existence of multiple dark states.

Emergent Chirality and Hyperuniformity in an Active Mixture with Nonreciprocal Interactions

We investigate collective dynamics in a binary mixture of programmable robots in experiments and simulations. While robots of the same species align their motion direction, interaction between species is distinctly nonreciprocal: species A aligns with B and species B antialigns with A. This nonreciprocal interaction gives rise to the emergence of collective chiral motion that can be stabilized by limiting the robot angular speed to be below a threshold. Within the chiral phase, increasing the robot density or extending the range of local repulsive interactions can drive the system through an absorbing-active transition. At the transition point, the robots exhibit a remarkable capacity for self-organization, forming disordered hyperuniform states.

Discontinuous phase transition from ferromagnetic to oscillating states in a nonequilibrium mean-field spin model

We study a nonequilibrium ferromagnetic mean-field spin model exhibiting a phase with spontaneous temporal oscillations of the magnetization, on top of the usual paramagnetic and ferromagnetic phases. This behavior is obtained by introducing dynamic field variables coupled to the spins through nonreciprocal couplings. We determine a nonequilibrium generalization of the Landau free energy in terms of the large deviation function of the magnetization and of an appropriately defined smoothed stochastic time derivative of the magnetization. While the transition between paramagnetic and oscillating phase is continuous, the transition between ferromagnetic and oscillating phases is found to be discontinuous, with a coexistence of both phases, one being stable and the other one metastable. Depending on parameter values, the ferromagnetic points may either be inside or outside the limit cycle, leading to different transition scenarios. The stability of these steady states is determined from the large deviation function. We also show that in the coexistence region, the entropy production has a pronounced maximum as a function of system size.

Irreversibility across a Nonreciprocal PT-Symmetry-Breaking Phase Transition

Nonreciprocal interactions are commonplace in continuum-level descriptions of both biological and synthetic active matter, yet studies addressing their implications for time reversibility have so far been limited to microscopic models. Here, we derive a general expression for the average rate of informational entropy production in the most generic mixture of conserved phase fields with nonreciprocal couplings and additive conservative noise. For the particular case of a binary system with Cahn-Hilliard dynamics augmented by nonreciprocal cross-diffusion terms, we observe a nontrivial scaling of the entropy production rate across a parity-time symmetry breaking phase transition. We derive a closed-form analytic expression in the weak-noise regime for the entropy production rate due to the emergence of a macroscopic dynamic phase, showing it can be written in terms of the global polar order parameter, a measure of parity-time symmetry breaking.

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases

We study fluctuating field models with spontaneously emerging dynamical phases. We consider two typical transition scenarios associated with parity-time symmetry breaking: oscillatory instabilities and critical exceptional points. An analytical investigation of the low-noise regime reveals a drastic increase of the mesoscopic entropy production toward the transitions. For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields, we find physical interpretations in terms of actively propelled interfaces and a coupling of eigenmodes of the linearized dynamics near the critical exceptional point.

Time-reversal and parity-time symmetry breaking in non-Hermitian field theories

We study time-reversal symmetry breaking in non-Hermitian fluctuating field theories with conserved dynamics, comprising the mesoscopic descriptions of a wide range of nonequilibrium phenomena. They exhibit continuous parity-time (PT) symmetry-breaking phase transitions to dynamical phases. For two concrete transition scenarios, exclusive to non-Hermitian dynamics, namely, oscillatory instabilities and critical exceptional points, a low-noise expansion exposes a pretransitional surge of the mesoscale (informatic) entropy production rate, inside the static phases. Its scaling in the susceptibility contrasts conventional critical points (such as second-order phase transitions), where the susceptibility also diverges, but the entropy production generally remains finite. The difference can be attributed to active fluctuations in the wavelengths that become unstable. For critical exceptional points, we identify the coupling of eigenmodes as the entropy-generating mechanism, causing a drastic noise amplification in the Goldstone mode.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

Entropy production in the nonreciprocal Cahn-Hilliard model

We study the nonreciprocal Cahn-Hilliard model with thermal noise as a prototypical example of a generic class of non-Hermitian stochastic field theories, analyzed in two companion papers [Suchanek, Kroy, and Loos, Phys. Rev. Lett. 131, 258302 (2023)10.1103/PhysRevLett.131.258302; Phys. Rev. E 108, 064123 (2023)10.1103/PhysRevE.108.064123]. Due to the nonreciprocal coupling between two field components, the model is inherently out of equilibrium and can be regarded as an active field theory. Beyond the conventional homogeneous and static-demixed phases, it exhibits a traveling-wave phase, which can be entered via either an oscillatory instability or a critical exceptional point. By means of a Fourier decomposition of the entropy production rate, we quantify the associated scale-resolved time-reversal symmetry breaking, in all phases and across the transitions, in the low-noise regime. Our perturbative calculation reveals its dependence on the strength of the nonreciprocal coupling. Surging entropy production near the static-dynamic transitions can be attributed to entropy-generating fluctuations in the longest wavelength Fourier mode and heralds the emerging traveling wave. Its translational dynamics can be mapped on the dissipative ballistic motion of an active (quasi)particle.

Brownian motors powered by nonreciprocal interactions

Traditional models for molecular (Brownian) motors predominantly depend on nonequilibrium driving, while particle interactions rigorously adhere to Newton's third law. However, numerous living and natural systems at various scales seem to defy this well-established law. In this study, we investigated the transport of mixed Brownian particles in a two-dimensional ratchet potential with nonreciprocal interactions. Our findings reveal that these nonreciprocal interactions can introduce a zero-mean nonequilibrium driving force. This force is capable of disrupting the thermodynamic equilibrium and inducing directed motion. The direction of this motion is determined by the asymmetry of the potential. Interestingly, the average velocity is a peaked function of the degree of nonreciprocity, while the effective diffusion consistently increases with the increase of nonreciprocity. There exists an optimal temperature or packing fraction at which the average velocity reaches its maximum value. We share a mechanism for particle rectification, devoid of particle-autonomous nonequilibrium drive, with potential usage in systems characterized by nonreciprocal interactions.

Non-reciprocity across scales in active mixtures

In active matter, particles typically experience mediated interactions, which are not constrained by Newton's third law and are therefore generically non-reciprocal. Non-reciprocity leads to a rich set of emerging behaviors that are hard to account for starting from the microscopic scale, due to the absence of a generic theoretical framework out of equilibrium. Here we consider bacterial mixtures that interact via mediated, non-reciprocal interactions (NRI) like quorum-sensing and chemotaxis. By explicitly relating microscopic and macroscopic dynamics, we show that, under conditions that we derive explicitly, non-reciprocity may fade upon coarse-graining, leading to large-scale equilibrium descriptions. In turn, this allows us to account quantitatively, and without fitting parameters, for the rich behaviors observed in microscopic simulations including phase separation, demixing, and multi-phase coexistence. We also derive the condition under which non-reciprocity survives coarse-graining, leading to a wealth of dynamical patterns. Again, our analytical approach allows us to predict the phase diagram of the system starting from its microscopic description. All in all, our work demonstrates that the fate of non-reciprocity across scales is a subtle and important question.

Enhanced diffusion of tracer particles in nonreciprocal mixtures

We study the diffusivity of a tagged particle in a binary mixture of Brownian particles with nonreciprocal interactions. Numerical simulations reveal that, for a broad class of interaction potentials, nonreciprocity can significantly increase the long-time diffusion coefficient of tracer particles and that this diffusion enhancement is associated with a breakdown of the Einstein relation. These observations are quantified and confirmed via two different and complementary analytical approaches: (i) a linearized stochastic density field theory, which is particularly accurate in the limit of soft interactions, and (ii) a reduced two-body description, which is exact at leading order in the density of particles. The latter reveals that diffusion enhancement can be attributed to the formation of transiently propelled dimers of particles, whose cohesion and speed are controlled by the nonreciprocal interactions.

Method of spectral response to stochastic processes for measuring the nonreciprocal effective interactions

The theoretical background of the nonperturbative method of spectral response to stochastic processes (SRSP) for measuring the nonreciprocal interparticle effective interactions in strongly coupled underdamped systems is described. Analytical expressions for vibrational spectral density of confined Brownian particles with a nonreciprocal effective interaction are presented. The changes in the vibrational spectral density with varying different parameters of the system (nonreciprocity, viscosity, ratios of particle sizes, and intensities of random processes acting on each particle) are discussed using the example of a pair of nonidentical particles in a harmonic trap. The SRSP method is compared to three other nonperturbative methods. The SRSP method demonstrates an undeniable advantage when processing particle trajectories with errors in particle tracking.

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 nonreciprocal 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 multicomponent active systems.

Dynamical Pattern Formation without Self-Attraction in Quorum-Sensing Active Matter: The Interplay between Nonreciprocity and Motility

We study a minimal model involving two species of particles interacting via quorum-sensing rules. Combining simulations of the microscopic model and linear stability analysis of the associated coarse-grained field theory, we identify a mechanism for dynamical pattern formation that does not rely on the standard route of intraspecies effective attractive interactions. Instead, our results reveal a highly dynamical phase of chasing bands induced only by the combined effects of self-propulsion and nonreciprocity in the interspecies couplings. Turning on self-attraction, we find that the system may phase separate into a macroscopic domain of such chaotic chasing bands coexisting with a dilute gas. We show that the chaotic dynamics of bands at the interfaces of this phase-separated phase results in anomalously slow coarsening.

Pattern formation by bacteria-phage interactions

The interactions between bacteria and phages-viruses that infect bacteria-play critical roles in agriculture, ecology, and medicine; however, how these interactions influence the spatial organization of both bacteria and phages remain largely unexplored. Here, we address this gap in knowledge by developing a theoretical model of motile, proliferating bacteria that aggregate via motility-induced phase separation (MIPS) and encounter phage that infect and lyse the cells. We find that the non-reciprocal predator-prey interactions between phage and bacteria strongly alter spatial organization, in some cases giving rise to a rich array of finite-scale stationary and dynamic patterns in which bacteria and phage coexist. We establish principles describing the onset and characteristics of these diverse behaviors, thereby helping to provide a biophysical basis for understanding pattern formation in bacteria-phage systems, as well as in a broader range of active and living systems with similar predator-prey or other non-reciprocal interactions.

Chemotactic Motility-Induced Phase Separation

Collectives of actively moving particles can spontaneously separate into dilute and dense phases-a fascinating phenomenon known as motility-induced phase separation (MIPS). MIPS is well-studied for randomly moving particles with no directional bias. However, many forms of active matter exhibit collective chemotaxis, directed motion along a chemical gradient that the constituent particles can generate themselves. Here, using theory and simulations, we demonstrate that collective chemotaxis strongly competes with MIPS-in some cases, arresting or completely suppressing phase separation, or in other cases, generating fundamentally new dynamic instabilities. We establish principles describing this competition, thereby helping to reveal and clarify the rich physics underlying active matter systems that perform chemotaxis, ranging from cells to robots.

Nonreciprocal Cahn-Hilliard Model Emerges as a Universal Amplitude Equation

Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatiotemporal pattern formation. Starting from a linear large-scale oscillatory instability-a conserved-Hopf instability-that naturally occurs in many active systems with two conservation laws, we derive a corresponding amplitude equation. It belongs to a hierarchy of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s). The derived universal equation generalizes a phenomenological model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard model, and may be of a similar relevance for the classification of pattern forming systems as the complex Ginzburg-Landau equation.

On-Demand Breaking of Action-Reaction Reciprocity between Magnetic Microdisks Using Global Stimuli

Coupled physical interactions induce emergent collective behaviors of many interacting objects. Nonreciprocity in the interactions generates unexpected behaviors. There is a lack of experimental model system that switches between the reciprocal and nonreciprocal regime on demand. Here, we study a system of magnetic microdisks that breaks action-reaction reciprocity via fluid-mediated hydrodynamic interactions, on demand. Via experiments and simulations, we demonstrate that nonreciprocal interactions generate self-propulsion-like behaviors of a pair of disks; group separation in collective of magnetically nonidentical disks; and decouples a part of the group from the rest. Our results could help in developing controllable microrobot collectives. Our approach highlights the effect of global stimuli in generating nonreciprocal interactions.

Self-organization of primitive metabolic cycles due to non-reciprocal interactions

One of the greatest mysteries concerning the origin of life is how it has emerged so quickly after the formation of the earth. In particular, it is not understood how metabolic cycles, which power the non-equilibrium activity of cells, have come into existence in the first instances. While it is generally expected that non-equilibrium conditions would have been necessary for the formation of primitive metabolic structures, the focus has so far been on externally imposed non-equilibrium conditions, such as temperature or proton gradients. Here, we propose an alternative paradigm in which naturally occurring non-reciprocal interactions between catalysts that can partner together in a cyclic reaction lead to their recruitment into self-organized functional structures. We uncover different classes of self-organized cycles that form through exponentially rapid coarsening processes, depending on the parity of the cycle and the nature of the interaction motifs, which are all generic but have readily tuneable features.

Stationary broken parity states in active matter models

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.

Nonequilibrium Phase Transition to Temporal Oscillations in Mean-Field Spin Models

We propose a mean-field theory to describe the nonequilibrium phase transition to a spontaneously oscillating state in spin models. A nonequilibrium generalization of the Landau free energy is obtained from the joint distribution of the magnetization and its smoothed stochastic time derivative. The order parameter of the transition is a Hamiltonian, whose nonzero value signals the onset of oscillations. The Hamiltonian and the nonequilibrium Landau free energy are determined explicitly from the stochastic spin dynamics. The oscillating phase is also characterized by a nontrivial overlap distribution reminiscent of a continuous replica symmetry breaking, in spite of the absence of disorder. An illustration is given on an explicit kinetic mean-field spin model.

Long-Range Order and Directional Defect Propagation in the Nonreciprocal XY Model with Vision Cone Interactions

We study a two-dimensional, nonreciprocal XY model, where each spin interacts only with its nearest neighbors in a certain angle around its current orientation, i.e., its "vision cone." Using energetic arguments and Monte Carlo simulations, we show that a true long-range ordered phase emerges. A necessary ingredient is a configuration-dependent bond dilution entailed by the vision cones. Strikingly, defects propagate in a directional manner, thereby breaking the parity and time-reversal symmetry of the spin dynamics. This is detectable by a nonzero entropy production rate.

Mechanical theory of nonequilibrium coexistence and motility-induced phase separation

Nonequilibrium phase transitions are routinely observed in both natural and synthetic systems. The ubiquity of these transitions highlights the conspicuous absence of a general theory of phase coexistence that is broadly applicable to both nonequilibrium and equilibrium systems. Here, we present a general mechanical theory for phase separation rooted in ideas explored nearly a half-century ago in the study of inhomogeneous fluids. The core idea is that the mechanical forces within the interface separating two coexisting phases uniquely determine coexistence criteria, regardless of whether a system is in equilibrium or not. We demonstrate the power and utility of this theory by applying it to active Brownian particles, predicting a quantitative phase diagram for motility-induced phase separation in both two and three dimensions. This formulation additionally allows for the prediction of novel interfacial phenomena, such as an increasing interface width while moving deeper into the two-phase region, a uniquely nonequilibrium effect confirmed by computer simulations. The self-consistent determination of bulk phase behavior and interfacial phenomena offered by this mechanical perspective provide a concrete path forward toward a general theory for nonequilibrium phase transitions.

Nonequilibrium currents in stochastic field theories: A geometric insight

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Non-reciprocity induces resonances in a two-field Cahn-Hilliard model

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Nonreciprocal forces enable cold-to-hot heat transfer between nanoparticles

We study the heat transfer between two nanoparticles held at different temperatures that interact through nonreciprocal forces, by combining molecular dynamics simulations with stochastic thermodynamics. Our simulations reveal that it is possible to construct nano refrigerators that generate a net heat transfer from a cold to a hot reservoir at the expense of power exerted by the nonreciprocal forces. Applying concepts from stochastic thermodynamics to a minimal underdamped Langevin model, we derive exact analytical expressions predictions for the fluctuations of work, heat, and efficiency, which reproduce thermodynamic quantities extracted from the molecular dynamics simulations. The theory only involves a single unknown parameter, namely an effective friction coefficient, which we estimate fitting the results of the molecular dynamics simulation to our theoretical predictions. Using this framework, we also establish design principles which identify the minimal amount of entropy production that is needed to achieve a certain amount of uncertainty in the power fluctuations of our nano refrigerator. Taken together, our results shed light on how the direction and fluctuations of heat flows in natural and artificial nano machines can be accurately quantified and controlled by using nonreciprocal forces.

Non-reciprocal multifarious self-organization

A hallmark of living systems is the ability to employ a common set of building blocks that can self-organize into a multitude of different structures. This capability can only be afforded in non-equilibrium conditions, as evident from the energy-consuming nature of the plethora of such dynamical processes. To achieve automated dynamical control of such self-assembled structures and transitions between them, we need to identify the fundamental aspects of non-equilibrium dynamics that can enable such processes. Here we identify programmable non-reciprocal interactions as a tool to achieve such functionalities. The design rule is composed of reciprocal interactions that lead to the equilibrium assembly of the different structures, through a process denoted as multifarious self-assembly, and non-reciprocal interactions that give rise to non-equilibrium dynamical transitions between the structures. The design of such self-organized shape-shifting structures can be implemented at different scales, from nucleic acids and peptides to proteins and colloids.

Catalysis-Induced Phase Separation and Autoregulation of Enzymatic Activity

We present a thermodynamically consistent model describing the dynamics of a multicomponent mixture where one enzyme component catalyzes a reaction between other components. We find that the catalytic activity alone can induce phase separation for sufficiently active systems and large enzymes, without any equilibrium interactions between components. In the limit of fast reaction rates, binodal lines can be calculated using a mapping to an effective free energy. We also explain how this catalysis-induced phase separation can act to autoregulate the enzymatic activity, which points at the biological relevance of this phenomenon.

Unjamming and emergent nonreciprocity in active ploughing through a compressible viscoelastic fluid

A dilute suspension of active Brownian particles in a dense compressible viscoelastic fluid, forms a natural setting to study the emergence of nonreciprocity during a dynamical phase transition. At these densities, the transport of active particles is strongly influenced by the passive medium and shows a dynamical jamming transition as a function of activity and medium density. In the process, the compressible medium is actively churned up - for low activity, the active particle gets self-trapped in a cavity of its own making, while for large activity, the active particle ploughs through the medium, either accompanied by a moving anisotropic wake, or leaving a porous trail. A hydrodynamic approach makes it evident that the active particle generates a long-range density wake which breaks fore-aft symmetry, consistent with the simulations. Accounting for the back-reaction of the compressible medium leads to (i) dynamical jamming of the active particle, and (ii) a dynamical non-reciprocal attraction between two active particles moving along the same direction, with the trailing particle catching up with the leading one in finite time. We emphasize that these nonreciprocal effects appear only when the active particles are moving and so manifest in the vicinity of the jamming-unjamming transition.

Active nonreciprocal attraction between motile particles in an elastic medium

We show from experiments and simulations on vibration-activated granular matter that self-propelled polar rods in an elastic medium on a substrate turn and move towards each other. We account for this effective attraction through a coarse-grained theory of a motile particle as a moving point-force density that creates elastic strains in the medium that reorient other particles. Our measurements confirm qualitatively the predicted features of the distortions created by the rods, including the |x|^{-1/2} tail of the trailing displacement field and nonreciprocal sensing and pursuit. A discrepancy between the magnitudes of displacements along and transverse to the direction of motion remains. Our theory should be of relevance to the interaction of motile cells in the extracellular matrix or in a supported layer of gel or tissue.

A topological fluctuation theorem

Fluctuation theorems specify the non-zero probability to observe negative entropy production, contrary to a naive expectation from the second law of thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can be used to give a geometric characterization of the entropy production. Building on this picture, we formulate a topological fluctuation theorem that depends only by the winding number around each vortex core and is insensitive to other aspects of the force. The probability is robust to local deformations of the particle trajectory, reminiscent of topologically protected modes in various classical and quantum systems. We demonstrate that entropy production is quantized in these strongly fluctuating systems, and it is controlled by a topological invariant. We demonstrate that the theorem holds even when the probability distributions are non-Gaussian functions of the generated heat.

While topology is crucial in complex systems, stochastic thermodynamics uncovers universal constraints for non-equilibrium fluctuations. The authors combine these two areas and formulate a fluctuation theorem for the heat dissipated along closed loops in vortex force fields, which is found to be topologically protected.

Realization of active metamaterials with odd micropolar elasticity

Materials made from active, living, or robotic components can display emergent properties arising from local sensing and computation. Here, we realize a freestanding active metabeam with piezoelectric elements and electronic feed-forward control that gives rise to an odd micropolar elasticity absent in energy-conserving media. The non-reciprocal odd modulus enables bending and shearing cycles that convert electrical energy into mechanical work, and vice versa. The sign of this elastic modulus is linked to a non-Hermitian topological index that determines the localization of vibrational modes to sample boundaries. At finite frequency, we can also tune the phase angle of the active modulus to produce a direction-dependent bending modulus and control non-Hermitian vibrational properties. Our continuum approach, built on symmetries and conservation laws, could be exploited to design others systems such as synthetic biofilaments and membranes with feed-forward control loops.

Non-equilibrium phase separation in mixtures of catalytically active particles: size dispersity and screening effects

Biomolecular condensates in cells are often rich in catalytically active enzymes. This is particularly true in the case of the large enzymatic complexes known as metabolons, which contain different enzymes that participate in the same catalytic pathway. One possible explanation for this self-organization is the combination of the catalytic activity of the enzymes and a chemotactic response to gradients of their substrate, which leads to a substrate-mediated effective interaction between enzymes. These interactions constitute a purely non-equilibrium effect and show exotic features such as non-reciprocity. Here, we analytically study a model describing the phase separation of a mixture of such catalytically active particles. We show that a Michaelis-Menten-like dependence of the particles' activities manifests itself as a screening of the interactions, and that a mixture of two differently sized active species can exhibit phase separation with transient oscillations. We also derive a rich stability phase diagram for a mixture of two species with both concentration-dependent activity and size dispersity. This work highlights the variety of possible phase separation behaviours in mixtures of chemically active particles, which provides an alternative pathway to the passive interactions more commonly associated with phase separation in cells. Our results highlight non-equilibrium organizing principles that can be important for biologically relevant liquid-liquid phase separation.

Long-Range Nematic Order in Two-Dimensional Active Matter

Working in two space dimensions, we show that the orientational order emerging from self-propelled polar particles aligning nematically is quasi-long-ranged beyond ℓ_{r}, the scale associated to induced velocity reversals, which is typically extremely large and often cannot even be measured. Below ℓ_{r}, nematic order is long-range. We construct and study a hydrodynamic theory for this de facto phase and show that its structure and symmetries differ from conventional descriptions of active nematics. We check numerically our theoretical predictions, in particular the presence of π-symmetric propagative sound modes, and provide estimates of all scaling exponents governing long-range space-time correlations.

Layered Chiral Active Matter: Beyond Odd Elasticity

In equilibrium liquid crystals, chirality leads to a variety of spectacular three-dimensional structures, but chiral and achiral phases with the same broken continuous symmetries have identical long-time, large-scale dynamics. In this Letter, starting from active model H^{*}, the general hydrodynamics of a pseudoscalar in a momentum-conserving fluid, we demonstrate that chirality qualitatively modifies the dynamics of layered liquid crystals in active systems in both two and three dimensions due to an active "odder" elasticity. In three dimensions, we demonstrate that the hydrodynamics of active cholesterics differs fundamentally from smectic-A liquid crystals, unlike their equilibrium counterpart. This distinction can be used to engineer a columnar array of vortices, with an antiferromagnetic vorticity alignment, that can be switched on and off by external strain. A two-dimensional chiral layered state-an array of lines on an incompressible, freestanding film of chiral active fluid with a preferred normal direction-is generically unstable. However, this instability can be tuned in easily realizable experimental settings when the film is either on a substrate or in an ambient fluid.

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.