Active Matter Systems Research Articles

Emergent behaviors of buckling-driven elasto-active structures

Active systems of self-propelled agents, e.g., birds, fish, and bacteria, can organize their collective motion into myriad autonomous behaviors. Ubiquitous in nature and across length scales, such phenomena are also amenable to artificial settings, e.g., where brainless self-propelled robots orchestrate their movements into spatial-temporal patterns via the application of external cues or when confined within flexible boundaries. Like their natural counterparts, these approaches typically require many units to initiate collective motion, so controlling the ensuing dynamics is challenging. Here, we demonstrate a simple mechanism that leverages nonlinear elasticity to tame near-diffusive motile particles in forming structures capable of directed motion and other emergent behaviors. Our elasto-active system comprises two centimeter-sized self-propelled microbots connected with elastic beams. These microbots exert forces that suffice to buckle the beam and set the structure in motion. We first rationalize the physics of the interaction between the beam and the microbots. Then we use reduced-order models to predict the interactions of our elasto-active structures with boundaries, e.g., walls and constrictions, and demonstrate how they can exhibit remarkable emergent behaviors such as maze navigation. These findings demonstrate that allowing and understanding changes in body morphology can enhance the capabilities of active matter systems and enable the design of robotic materials capable of space exploration, adaptation, and complex interactions with their surrounding environment.

Tuning collective behaviour in zebrafish with genetic modification.

Zebrafish collective behaviour is widely used to assess their physical and mental state, serving as a valuable tool to assess the impact of ageing, disease genetics, and the effect of drugs. The essence of these macroscopic phenomena can be represented by active matter models, where the individuals are abstracted as interactive self-propelling agents. The behaviour of these agents depends on a set of parameters in a manner reminiscent of those between the constituents of physical systems. In a few cases, the system may be controlled at the level of the individual constituents such as the interactions between colloidal particles, or the enzymatic behaviour of de novo proteins. Usually, however, while the collective behaviour may be influenced by environmental factors, it typically cannot be changed at will. Here, we challenge this scenario in a biological context by genetically modifying zebrafish. We thus demonstrate the potential of genetic modification in the context of controlling the collective behaviour of biological active matter systems at the level of the constituents, rather than externally. In particular, we probe the effect of the lack of col11a2 gene in zebrafish, which causes the early onset of osteoarthritis. The resulting col11a2 -/- zebrafish exhibited compromised vertebral column properties, bent their body less while swimming, and took longer to change their orientations. Surprisingly, a group of 25 mutant fish exhibited more orderly collective motion than the wildtype. We show that the collective behaviour of wildtype and col11a2 -/- zebrafish are captured with a simple active matter model, in which the mutant fish are modelled by self-propelling agents with a higher orientational noise on average. In this way, we demonstrate the possibility of tuning a biological system, changing the state space it occupies when interpreted with a simple active matter model.

Learning protocols for the fast and efficient control of active matter

Exact analytic calculation shows that optimal control protocols for passive molecular systems often involve rapid variations and discontinuities. However, similar analytic baselines are not generally available for active-matter systems, because it is more difficult to treat active systems exactly. Here we use machine learning to derive efficient control protocols for active-matter systems, and find that they are characterized by sharp features similar to those seen in passive systems. We show that it is possible to learn protocols that effect fast and efficient state-to-state transformations in simulation models of active particles by encoding the protocol in the form of a neural network. We use evolutionary methods to identify protocols that take active particles from one steady state to another, as quickly as possible or with as little energy expended as possible. Our results show that protocols identified by a flexible neural-network ansatz, which allows the optimization of multiple control parameters and the emergence of sharp features, are more efficient than protocols derived recently by constrained analytical methods. Our learning scheme is straightforward to use in experiment, suggesting a way of designing protocols for the efficient manipulation of active matter in the laboratory.

Dynamical clustering and wetting phenomena in inertial active matter

Dynamical clustering is a key feature of active matter systems composed of self-propelled agents that convert environmental energy into mechanical motion. At the micron scale, where overdamped dynamics dominate, particles with opposite motility can obstruct each other’s movement, leading to transient dynamical arrest. This arrest can promote cluster formation and motility-induced phase separation. However, in macroscopic agents, where inertia plays a significant role, clustering is heavily influenced by bounce-back effects during collisions, which can impede cluster growth. Here we present an experiment based on active granular particles, in which inertia can be systematically tuned by changing the shaker frequency. As a result, a set of phenomena driven and controlled by inertia emerges. Before the suppression of clustering, inertia induces a transition in the cluster’s inner structure. For small inertia, clusters are characterized by the crystalline order typical of overdamped particles, while for large inertia clusters with liquid-like order are observed. In addition, in contrast to microswimmers, where active particles wet the boundary by primarily forming clusters attached to the container walls, in an underdamped inertial active system, walls do not favor cluster formation and effectively annihilate motility-induced wetting phenomena. As a consequence, inertia suppresses cluster nucleation at the system boundaries.

Machine learning assisted sorting of active microswimmers.

Active matter systems, being in a non-equilibrium state, exhibit complex behaviors, such as self-organization, giving rise to emergent phenomena. There are many examples of active particles with biological origins, including bacteria and spermatozoa, or with artificial origins, such as self-propelled swimmers and Janus particles. The ability to manipulate active particles is vital for their effective application, e.g., separating motile spermatozoa from nonmotile and dead ones, to increase fertilization chance. In this study, we proposed a mechanism-an apparatus-to sort and demix active particles based on their motility values (Péclet number). Initially, using Brownian simulations, we demonstrated the feasibility of sorting self-propelled particles. Following this, we employed machine learning methods, supplemented with data from comprehensive simulations that we conducted for this study, to model the complex behavior of active particles. This enabled us to sort them based on their Péclet number. Finally, we evaluated the performance of the developed models and showed their effectiveness in demixing and sorting the active particles. Our findings can find applications in various fields, including physics, biology, and biomedical science, where the sorting and manipulation of active particles play a pivotal role.

Arbitrage equilibria in active matter systems

The motility-induced phase separation (MIPS) phenomenon in active matter has been of great interest for the past decade or so. A central conceptual puzzle is that this behavior, which is generally characterized as a nonequilibrium phenomenon, can yet be explained using simple equilibrium models of thermodynamics. Here, we address this problem using a new theory, statistical teleodynamics, which is a conceptual synthesis of game theory and statistical mechanics. In this framework, active agents compete in their pursuit of maximum effective utility, and this self-organizing dynamics results in an arbitrage equilibrium in which all agents have the same effective utility. We show that MIPS is an example of arbitrage equilibrium and that it is mathematically equivalent to other phase-separation phenomena in entirely different domains, such as sociology and economics. As examples, we present the behavior of Janus particles in a potential trap and the effect of chemotaxis on MIPS.

Control of colloidal cohesive states in active chiral fluids

Ensembles of suspended spinning particles in liquids form a distinct category of active matter systems known as chiral fluids. Recent experimental instances of dense chiral fluids have comprised of spinning colloidal magnets powered by an external rotating magnetic field. These particles interact through both magnetic and hydrodynamic forces, organizing collectively into circulating clusters characterized by unidirectional edge flows. Here, we externally drive the collective behavior of spinning colloids by leveraging diffusiophoretic interactions among the geometrically anisotropic particles. We show that these nanoscale interfacial flows lead to the formation of bound states between spinning colloids that are stabilized through near-field hydrodynamic and chemical interactions. At a collective level, we demonstrate that added diffusiophoretic interactions cause a loss in structural cohesion of the circulating clusters and promote expansion, while preserving global cluster inter-connectivity. The expanded cluster state is characterized by the formation of a dynamic interconnected network promoted by axi-asymmetric interactions around particles with attractive dipolar interactions dominating along the direction of the magnetic moment. This process is observed to be entirely reversible, offering external control over the emergent dynamics in dense chiral fluids, paving the way for new self-organization routes in chiral fluids and broader forms of active matter.

Long-range velocity correlations from active dopants

Active matter systems display collective behaviors that are impossible in thermodynamic equilibrium. One such feature, observed in in dense active matter systems is the appearance of long-range velocity correlations without explicit aligning interaction. However, the conditions for the appearance of these correlations remain largely unexplored. Here we show that such long-range velocity correlations can also be generated in a dense athermal passive system by the inclusion of a very small fraction of active Brownian particles. We develop a continuum theory to explain the emergence of velocity correlations generated via such active dopants. We validate the predictions for the effects of magnitude and persistence time of the active force and the area fractions of active and passive particles using extensive Brownian dynamics simulation of a canonical active-passive mixture. Our work decouples the roles that density and activity play in generating long-range velocity correlations in such exotic non-equilibrium steady states.

Simulating pedestrian avoidance: The human-zombie game.

This study introduces a simulated active matter system, applying the pedestrian collision avoidance paradigm, which involves dynamically adjusting the desired velocity. We present a human-zombie game set within a closed geometry, combining predator-prey behavior with a one-way contagion process that transforms prey into predators. The system demonstrates varied responses in our implemented model: with agents having the same maximum speeds, a single zombie always captures a human, whereas two zombies never capture a single human agent. As the number of human agents increases, observables such as the final fraction of zombie agents and total conversion times exhibit significant changes in the system's behavior at intermediate density values. Most notably, there is evidence of a first-order phase transition when analyzing the mean population speed as an order parameter.

Supramolecular Assemblies in Active Motor-Filament Systems: Micelles, Bilayers, and Foams

Active matter systems evade the constraints of thermal equilibrium, leading to the emergence of intriguing collective behavior. A paradigmatic example is given by motor-filament mixtures, where the motion of motor proteins drives alignment and sliding interactions between filaments and their self-organization into macroscopic structures. After defining a microscopic model for these systems, we derive continuum equations, exhibiting the formation of active supramolecular assemblies such as micelles, bilayers, and foams. The transition between these structures is driven by a branching instability, which destabilizes the orientational order within the micelles, leading to the growth of bilayers at high microtubule densities. Additionally, we identify a fingering instability, modulating the shape of the micelle interface at high motor densities. We study the role of various mechanisms in these two instabilities, such as contractility, active splay, and anchoring, allowing for generalization beyond the system considered here. Published by the American Physical Society 2024

Noisy Circumnutations Facilitate Self-Organized Shade Avoidance in Sunflowers

Circumnutations are widespread in plants and typically associated with exploratory movements; however, a quantitative understanding of their role remains elusive. In this study we report, for the first time, the role of noisy circumnutations in facilitating an optimal growth pattern within a crowded group of mutually shading plants. We revisit the problem of self-organization observed for sunflowers, mediated by shade response interactions. Our analysis reveals that circumnutation movements conform to a bounded random walk characterized by a remarkably broad distribution of velocities, covering 3 orders of magnitude. In motile animal systems such wide distributions of movement velocities are frequently identified with enhancement of behavioral processes, suggesting that circumnutations may serve as a source of functional noise. To test our hypothesis, we developed a Langevin-type parsimonious model of interacting growing disks, informed by experiments, successfully capturing the characteristic dynamics of individual and multiple interacting plants. Employing our simulation framework we examine the role of circumnutations in the system, and find that the observed breadth of the velocity distribution represents a sharp transition in the force-noise ratio, conferring advantageous effects by facilitating exploration of potential configurations, leading to an optimized arrangement with minimal shading. These findings represent the first report of functional noise in plant movements and establish a theoretical foundation for investigating how plants navigate their environment by employing computational processes such as task-oriented processes, optimization, and active sensing. Since plants move by growing, space and time are coupled, and dynamics of self-organization lead to emergent 3D patterns. As such, this system provides conceptual insight for other interacting growth-driven systems such as fungal hyphae, neurons and self-growing robots, as well as active matter systems where agents interact with past trajectories of their counterparts, such as stigmergy in social insects. This foundational insight has implications in statistical physics, ecological dynamics, agriculture, and even swarm robotics. Published by the American Physical Society 2024

Orbits, Spirals, and Trapped States: Dynamics of a Phoretic Janus Particle in a Radial Concentration Gradient.

A long-standing goal in colloidal active matter is to understand how gradients in fuel concentration influence the motion of phoretic Janus particles. Here, we present a theoretical description of the motion of a spherical phoretic Janus particle in the presence of a radial gradient of the chemical solute driving self-propulsion. Radial gradients are a geometry relevant to many scenarios in active matter systems and naturally arise due to the presence of a point source or sink of fuel. We derive an analytical solution for the Janus particle's velocity and quantify the influence of the radial concentration gradient on the particle's trajectory. Compared to a phoretic Janus particle in a linear gradient in fuel concentration, we uncover a much richer set of dynamic behaviors including circular orbits and trapped stationary states. We identify the ratio of the phoretic mobilities between the two domains of the Janus particle as a central quantity in tuning their dynamics. Our results provide a path for developing optimum protocols for tuning the dynamics of phoretic Janus particles and mixing fluid at the microscale. In addition, this work suggests a method for quantifying the surface properties of phoretic Janus particles, which have proven to be challenging to probe experimentally.

Geometric frustration and pairing-order transition in confined bacterial vortices

Dense systems of active matter exhibit highly dynamic collective motion characterized by intermingled vortices, referred to as active turbulence. The interaction between these vortices is key to controlling turbulent dynamics, and a promising approach for revealing the rules governing their interaction is geometric confinement. In this study, we investigate the vortex-pairing patterns in confined bacterial suspensions as a model frustrated system in which a perfect antiferromagnetic state is prohibited. We find that three-body vortex interactions exhibit anomalous pairing-order transition from corotational vortex pairing to counterrotating patterns with frustration. Although an active matter system is in nonequilibrium, our theory based on bending energy accounts for significant features, including pattern transition and a shift of the transition point in frustrated systems. Moreover, the interplay between the chirality in collective motion and frustration in vortex pairing creates a collective rotational flow under the broad geometric conditions of a confined space. Our results show that frustrated vortex patterning promotes a geometric approach for arranging active turbulence in microfluidic systems. Published by the American Physical Society 2024

Rigid flocks, undulatory gaits, and chiral foldamers in a chemically active polymer

Active matter systems—such as a collection of active colloidal particles—operate far from equilibrium with complex inter-particle interactions that govern their collective dynamics. Predicting the collective dynamics of such systems may aid the design of self-shaping structures comprised of active colloidal units with a prescribed dynamical function. Here, using simulations and theory, we study the collective dynamics of a chain consisting of active Brownian particles with internal interactions via trail-mediated chemicals, connected by harmonic springs in two dimensions to obtain design principles for active colloidal molecules. We show that two-dimensional confinement and chemo-repulsive interactions between the freely-jointed particles lead to an emergent rigidity of the chain in the steady-state dynamics. In the chemo-attractive regime, the chain collapses into crystals that abruptly halt their motion. Further, in a chain consisting of a binary mixture of monomers, we show that non-reciprocal chemical affinities between distinct species give rise to novel phenomena, such as chiral molecules with tunable dynamics, sustained undulatory gaits and reversal of the direction of motion. Our results suggest a novel interpretation of the role of trail-mediated interactions, in addition to providing active self-assembly principles arising due to non-reciprocal interactions.

Pattern formation in odd viscoelastic fluids

Nonreciprocal interactions fueled by local energy consumption can be found in biological and synthetic active matter at scales where viscoelastic forces are important. Such systems can be described by “odd” viscoelasticity, which assumes fewer material symmetries than traditional theories. Here we study odd viscoelasticity analytically and using lattice Boltzmann simulations. We identify a pattern-forming instability which produces an oscillating array of fluid vortices, and we elucidate which features govern the growth rate, wavelength, and saturation of the vortices. Our observation of pattern formation through odd mechanical response can inform models of biological patterning and guide engineering of odd dynamics in soft active matter systems. Published by the American Physical Society 2024

Unveiling the Phase Diagram and Reaction Paths of the Active Model B with the Deep Minimum Action Method.

Nonequilibrium phase transitions are notably difficult to analyze because their mechanisms depend on the system's dynamics in a complex way due to the lack of time-reversal symmetry. To complicate matters, the system's steady-state distribution is unknown in general. Here, the phase diagram of the active Model B is computed with a deep neural network implementation of the geometric minimum action method (gMAM). This approach unveils the unconventional reaction paths and nucleation mechanism in dimensions 1, 2, and 3, by which the system switches between the homogeneous and inhomogeneous phases in the binodal region. Our main findings are (i)the mean time to escape the phase-separated state is (exponentially) extensive in the system size L, but it increases nonmonotonically with L in dimension 1; (ii)the mean time to escape the homogeneous state is always finite, in line with the recent work of Cates and Nardini [Phys. Rev. Lett. 130, 098203 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.098203]; (iii)at fixed L, the active term increases the stability of the homogeneous phase, eventually destroying the phase separation in the binodal for large but finite systems. Our results are particularly relevant for active matter systems in which the number of constituents hardly goes beyond 10^{7} and where finite-size effects matter.

Self-consistent sharp interface theory of active condensate dynamics

Biomolecular condensates help organize the cell cytoplasm and nucleoplasm into spatial compartments with different chemical compositions. A key feature of such compositional patterning is the local enrichment of enzymatically active biomolecules which, after transient binding via molecular interactions, catalyze reactions among their substrates. Thereby, biomolecular condensates provide a spatial template for nonuniform concentration profiles of substrates. In turn, the concentration profiles of substrates, and their molecular interactions with enzymes, drive enzyme fluxes which can enable novel nonequilibrium dynamics. To analyze this generic class of systems, with a current focus on self-propelled droplet motion, we here develop a self-consistent sharp interface theory. In our theory, we diverge from the usual bottom-up approach, which involves calculating the dynamics of concentration profiles based on a given chemical potential gradient. Instead, reminiscent of control theory, we take the reverse approach by deriving the chemical potential profile and enzyme fluxes required to maintain a desired condensate form and dynamics. The chemical potential profile and currents of enzymes come with a corresponding power dissipation rate, which allows us to derive a thermodynamic consistency criterion for the passive part of the system (here, reciprocal enzyme-enzyme interactions). As a first-use case of our theory, we study the role of reciprocal interactions, where the transport of substrates due to reactions and diffusion is, in part, compensated by redistribution due to molecular interactions. More generally, our theory applies to mass-conserved active matter systems with moving phase boundaries. Published by the American Physical Society 2024

Interplay of chemotactic force, Péclet number, and dimensionality dictates the dynamics of auto-chemotactic chiral active droplets.

In living and synthetic active matter systems, the constituents can self-propel and interact with each other and with the environment through various physicochemical mechanisms. Among these mechanisms, chemotactic and auto-chemotactic effects are widely observed. The impact of (auto-)chemotactic effects on achiral active matter has been a recent research focus. However, the influence of these effects on chiral active matter remains elusive. Here, we develop a Brownian dynamics model coupled with a diffusion equation to examine the dynamics of auto-chemotactic chiral active droplets in both quasi-two-dimensional (2D) and three-dimensional (3D) systems. By quantifying the droplet trajectory as a function of the dimensionless Péclet number and chemotactic strength, our simulations well reproduce the curling and helical trajectories of nematic droplets in a surfactant-rich solution reported by Krüger et al. [Phys. Rev. Lett. 117, 048003 (2016)]. The modeled curling trajectory in 2D exhibits an emergent chirality, also consistent with the experiment. We further show that the geometry of the chiral droplet trajectories, characterized by the pitch and diameter, can be used to infer the velocities of the droplet. Interestingly, we find that, unlike the achiral case, the velocities of chiral active droplets show dimensionality dependence: its mean instantaneous velocity is higher in 3D than in 2D, whereas its mean migration velocity is lower in 3D than in 2D. Taken together, our particle-based simulations provide new insights into the dynamics of auto-chemotactic chiral active droplets, reveal the effects of dimensionality, and pave the way toward their applications, such as drug delivery, sensors, and micro-reactors.

Nonlinear response theory of molecular machines

Chemical affinities are responsible for driving active matter systems out of equilibrium. At the nano-scale, molecular machines interact with the surrounding environment and are subjected to external forces. The mechano-chemical coupling which arises naturally in these systems reveals a complex interplay between chemical and mechanical degrees of freedom with strong impact on their active mechanism. By considering various models far from equilibrium, we show that the tuning of applied forces gives rise to a nonlinear response that causes a non-monotonic behaviour in the machines’ activity. Our findings have implications in understanding, designing, and triggering such processes by controlled application of external fields, including the collective dynamics of larger non-equilibrium systems where the total dissipation and performance might be affected by internal and inter-particle interactions.

Deep learning probability flows and entropy production rates in active matter

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a spatially local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4,096 particles at one packing fraction can generalize to other regions of the phase diagram, including to systems with as many as 32,768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.