Effective temperature of active matter

Dynamical crossovers and correlations in a harmonic chain of active particles

We explore the dynamics of a tracer in an active particle harmonic chain, investigating the influence of interactions. Our analysis involves calculating mean-squared displacements (MSDs) and space-time correlations through Green's function techniques and numerical simulations. Depending on chain characteristics, i.e., different time scales determined by interaction stiffness and persistence of activity, tagged-particle MSDs exhibit ballistic, diffusive, and single-file diffusion (SFD) scaling over time, with crossovers explained by our analytic expressions. Our results reveal transitions in bulk particle displacement distributions from an early-time bimodal to late-time Gaussian, passing through regimes of unimodal distributions with finite support and negative excess kurtosis and longer-tailed distributions with positive excess kurtosis. The distributions exhibit data collapse, aligning with ballistic, diffusive, and SFD scaling in the appropriate time regimes. However, at much longer times, the distributions become Gaussian. Finally, we derive analytic expressions for steady-state static and dynamic two-point displacement correlations. We verify these from simulations and highlight the differences from the equilibrium results.

Active fractal networks with stochastic force monopoles and force dipoles: Application to subdiffusion of chromosomal loci

Motivated by the well-known fractal packing of chromatin, we study the Rouse-type dynamics of elastic fractal networks with embedded, stochastically driven, active force monopoles and force dipoles that are temporally correlated. We compute, analytically-using a general theoretical framework-and via Langevin dynamics simulations, the mean square displacement (MSD) of a network bead. Following a short-time superdiffusive behavior, force monopoles yield anomalous subdiffusion with an exponent identical to that of the thermal system. In contrast, force dipoles do not induce subdiffusion, and the early superdiffusive MSD crosses over to a relatively small, system-size-independent saturation value. In addition, we find that force dipoles may lead to "crawling" rotational motion of the whole network, reminiscent of that found for triangular micro-swimmers and consistent with general theories of the rotation of deformable bodies. Moreover, force dipoles lead to network collapse beyond a critical force strength, which persists with increasing system size, signifying a true first-order dynamical phase transition. We apply our results to the motion of chromosomal loci in bacteria and yeast cells' chromatin, where anomalous sub-diffusion, MSD∼tν with ν≃0.4, was found in both normal and cells depleted of adenosine triphosphate (ATP), albeit with different apparent diffusion coefficients. We show that the combination of thermal, monopolar, and dipolar forces in chromatin is typically dominated by the active monopolar and thermal forces, explaining the observed normal cells vs the ATP-depleted cells behavior.

Mesoscale molecular assembly is favored by the active, crowded cytoplasm

The mesoscale organization of molecules into membraneless biomolecular condensates is emerging as a key mechanism of rapid spatiotemporal control in cells. Principles of biomolecular condensation have been revealed through in vitro reconstitution. However, intracellular environments are much more complex than test-tube environments: they are viscoelastic, highly crowded at the mesoscale, and are far from thermodynamic equilibrium due to the constant action of energy-consuming processes. We developed synDrops, a synthetic phase separation system, to study how the cellular environment affects condensate formation. Three key features enable physical analysis: synDrops are inducible, bioorthogonal, and have well-defined geometry. This design allows kinetic analysis of synDrop assembly and facilitates computational simulation of the process. We compared experiments and simulations to determine that macromolecular crowding promotes condensate nucleation but inhibits droplet growth through coalescence. ATP-dependent cellular activities help overcome the frustration of growth. In particular, stirring of the cytoplasm by actomyosin dynamics is the dominant mechanism that potentiates droplet growth in the mammalian cytoplasm by reducing confinement and elasticity. Our results demonstrate that mesoscale molecular assembly is favored by the combined effects of crowding and active matter in the cytoplasm. These results move toward a better predictive understanding of condensate formation in vivo.

Fluctuation Theorems for Heat Exchanges between Passive and Active Baths

In addition to providing general constraints on probability distributions, fluctuation theorems allow us to infer essential information on the role played by temperature in heat exchange phenomena. In this numerical study, we measure the temperature of an out-of-equilibrium active bath using a fluctuation theorem that relates the fluctuations in the heat exchanged between two baths to their temperatures. Our setup consists of a single particle moving between two wells of a quartic potential accommodating two different baths. The heat exchanged between the two baths is monitored according to two definitions: as the kinetic energy carried by the particle whenever it jumps from one well to the other and as the work performed by the particle on one of the two baths when immersed in it. First, we consider two equilibrium baths at two different temperatures and verify that a fluctuation theorem featuring the baths temperatures holds for both heat definitions. Then, we introduce an additional Gaussian coloured noise in one of the baths, so as to make it effectively an active (out-of-equilibrium) bath. We find that a fluctuation theorem is still satisfied with both heat definitions. Interestingly, in this case the temperature obtained through the fluctuation theorem for the active bath corresponds to the kinetic temperature when considering the first heat definition, while it is larger with the second one. We interpret these results by looking at the particle jump phenomenology.

Transcription-induced active forces suppress chromatin motion

The organization of interphase chromosomes in a number of species is starting to emerge thanks to advances in a variety of experimental techniques. However, much less is known about the dynamics, especially in the functional states of chromatin. Some experiments have shown that the motility of individual loci in human interphase chromosome decreases during transcription and increases upon inhibiting transcription. This is a counterintuitive finding because it is thought that the active mechanical force (F) on the order of ten piconewtons, generated by RNA polymerase II (RNAPII) that is presumably transmitted to the gene-rich region of the chromatin, would render it more open, thus enhancing the mobility. We developed a minimal active copolymer model for interphase chromosomes to investigate how F affects the dynamical properties of chromatin. The movements of the loci in the gene-rich region are suppressed in an intermediate range of F and are enhanced at small F values, which has also been observed in experiments. In the intermediate F, the bond length between consecutive loci increases, becoming commensurate with the distance at the minimum of the attractive interaction between nonbonded loci. This results in a transient disorder-to-order transition, leading to a decreased mobility during transcription. Strikingly, the F-dependent change in the locus dynamics preserves the organization of the chromosome at [Formula: see text]. Transient ordering of the loci, which is not found in the polymers with random epigenetic profiles, in the gene-rich region might be a plausible mechanism for nucleating a dynamic network involving transcription factors, RNAPII, and chromatin.

Coherent light scattering from cellular dynamics in living tissues

This review examines the biological physics of intracellular transport probed by the coherent optics of dynamic light scattering from optically thick living tissues. Cells and their constituents are in constant motion, composed of a broad range of speeds spanning many orders of magnitude that reflect the wide array of functions and mechanisms that maintain cellular health. From the organelle scale of tens of nanometers and upward in size, the motion inside living tissue is actively driven rather than thermal, propelled by the hydrolysis of bioenergetic molecules and the forces of molecular motors. Active transport can mimic the random walks of thermal Brownian motion, but mean-squared displacements are far from thermal equilibrium and can display anomalous diffusion through Lévy or fractional Brownian walks. Despite the average isotropic three-dimensional environment of cells and tissues, active cellular or intracellular transport of single light-scattering objects is often pseudo-one-dimensional, for instance as organelle displacement persists along cytoskeletal tracks or as membranes displace along the normal to cell surfaces, albeit isotropically oriented in three dimensions. Coherent light scattering is a natural tool to characterize such tissue dynamics because persistent directed transport induces Doppler shifts in the scattered light. The many frequency-shifted partial waves from the complex and dynamic media interfere to produce dynamic speckle that reveals tissue-scale processes through speckle contrast imaging and fluctuation spectroscopy. Low-coherence interferometry, dynamic optical coherence tomography, diffusing-wave spectroscopy, diffuse-correlation spectroscopy, differential dynamic microscopy and digital holography offer coherent detection methods that shed light on intracellular processes. In health-care applications, altered states of cellular health and disease display altered cellular motions that imprint on the statistical fluctuations of the scattered light. For instance, the efficacy of medical therapeutics can be monitored by measuring the changes they induce in the Doppler spectra of livingex vivocancer biopsies.

Effect of contact inhibition locomotion on confined cellular organization

Experiments performed using micro-patterned one dimensional collision assays have allowed a precise quantitative analysis of the collective manifestation of contact inhibition locomotion (CIL) wherein, individual migrating cells reorient their direction of motion when they come in contact with other cells. Inspired by these experiments, we present a discrete, minimal 1D Active spin model that mimics the CIL interaction between cells in one dimensional channels. We analyze the emergent collective behaviour of migrating cells in such confined geometries, as well as the sensitivity of the emergent patterns to driving forces that couple to cell motion. In the absence of vacancies, akin to dense cell packing, the translation dynamics is arrested and the model reduces to an equilibrium spin model which can be solved exactly. In the presence of vacancies, the interplay of activity-driven translation, cell polarity switching, and CIL results in an exponential steady cluster size distribution. We define a dimensionless Péclet number Q-the ratio of the translation rate and directional switching rate of particles in the absence of CIL. While the average cluster size increases monotonically as a function of Q, it exhibits a non-monotonic dependence on CIL strength, when the Q is sufficiently high. In the high Q limit, an analytical form of average cluster size can be obtained approximately by effectively mapping the system to an equivalent equilibrium process involving clusters of different sizes wherein the cluster size distribution is obtained by minimizing an effective Helmholtz free energy for the system. The resultant prediction of exponential dependence on CIL strength of the average cluster size and [Formula: see text] dependence of the average cluster size is borne out to reasonable accuracy as long as the CIL strength is not very large. The consequent prediction of a single scaling function of Q, particle density and CIL interaction strength, characterizing the distribution function of the cluster sizes and resultant data collapse is observed for a range of parameters.

How it feels in a cell

Life emerges from thousands of biochemical processes occurring within a shared intracellular environment. We have gained deep insights from in vitro reconstitution of isolated biochemical reactions. However, the reaction medium in test tubes is typically simple and diluted. The cell interior is far more complex: macromolecules occupy more than a third of the space, and energy-consuming processes agitate the cell interior. Here, we review how this crowded, active environment impacts the motion and assembly of macromolecules, with an emphasis on mesoscale particles (10-1000 nm diameter). We describe methods to probe and analyze the biophysical properties of cells and highlight how changes in these properties can impact physiology and signaling, and potentially contribute to aging, and diseases, including cancer and neurodegeneration.

Effective temperature for an intermittent bistable potential

Thermodynamics of far-from-equilibrium systems often require measurement of effective parameters such as temperature. Whether such approach is valid for the general case of resetting protocols, active systems, or of confined systems under time-varying fields is still under investigation. We report on the effect of switching ON-OFF of an asymmetric bistable potential to the mean first passage time (MFPT) of a probed particle to go from one potential minima to the other. Experimental results coupled with numerical simulations shows the potential becoming more symmetric at slow switching. Moreover, the MFPT deviates from equilibrium condition with an effective temperature, Teff < T, at slow switching but approaches room temperature, T, at fast switching. For each switching rate, we quantify how far the system is from equilibrium by measuring deviation from a detailed balance like relation and the net circulation of flux present in phase-space. Both analysis suggest equilibrium condition are met at high switching.

Mesoscale molecular assembly is favored by the active, crowded cytoplasm

The mesoscale organization of molecules into membraneless biomolecular condensates is emerging as a key mechanism of rapid spatiotemporal control in cells1. Principles of biomolecular condensation have been revealed through in vitro reconstitution2. However, intracellular environments are much more complex than test-tube environments: They are viscoelastic, highly crowded at the mesoscale, and are far from thermodynamic equilibrium due to the constant action of energy-consuming processes3. We developed synDrops, a synthetic phase separation system, to study how the cellular environment affects condensate formation. Three key features enable physical analysis: synDrops are inducible, bioorthogonal, and have well-defined geometry. This design allows kinetic analysis of synDrop assembly and facilitates computational simulation of the process. We compared experiments and simulations to determine that macromolecular crowding promotes condensate nucleation but inhibits droplet growth through coalescence. ATP-dependent cellular activities help overcome the frustration of growth. In particular, actomyosin dynamics potentiate droplet growth by reducing confinement and elasticity in the mammalian cytoplasm, thereby enabling synDrop coarsening. Our results demonstrate that mesoscale molecular assembly is favored by the combined effects of crowding and active matter in the cytoplasm. These results move toward a better predictive understanding of condensate formation in vivo.

Universal equation of state for wave turbulence in a quantum gas

Boyle's 1662 observation that the volume of a gas is, at constant temperature, inversely proportional to pressure, offered a prototypical example of how an equation of state (EoS) can succinctly capture key properties of a many-particle system. Such relationships are now cornerstones of equilibrium thermodynamics1. Extending thermodynamic concepts to far-from-equilibrium systems is of great interest in various contexts, including glasses2,3, active matter4-7 and turbulence8-11, but is in general an open problem. Here, using a homogeneous ultracold atomic Bose gas12, we experimentally construct an EoS for a turbulent cascade of matter waves13,14. Under continuous forcing at a large length scale and dissipation at a small one, the gas exhibits a non-thermal, but stationary, state, which is characterized by a power-law momentum distribution15 sustained by a scale-invariant momentum-space energy flux16. We establish the amplitude of the momentum distribution and the underlying energy flux as equilibrium-like state variables, related by an EoS that does not depend on the details of the energy injection or dissipation, or on the history of the system. Moreover, we show that the equations of state for a wide range of interaction strengths and gas densities can be empirically scaled onto each other. This results in a universal dimensionless EoS that sets benchmarks for the theory and should also be relevant for other turbulent systems.

Detecting and characterizing phase transitions in active matter using entropy

A major challenge in the study of active matter lies in quantitative characterization of phases and transitions between them. We show how the entropy of a collection of active objects can be used to classify regimes and spatial patterns in their collective behavior. Specifically, we estimate the contributions to the total entropy from correlations between the degrees of freedom of position and orientation. This analysis pin-points the flocking transition in the Vicsek model while clarifying the physical mechanism behind the transition. When applied to experiments on swarming Bacillus subtilis with different cell aspect ratios and overall bacterial area fractions, the entropy analysis reveals a rich phase diagram with transitions between qualitatively different swarm statistics. We discuss physical and biological implications of these findings.

A bending rigidity parameter for stress granule condensates

Interfacial tension plays an important role in governing the dynamics of droplet coalescence and determining how condensates interact with and deform lipid membranes and biological filaments. We demonstrate that an interfacial tension-only model is inadequate for describing stress granules in live cells. Harnessing a high-throughput flicker spectroscopy pipeline to analyze the shape fluctuations of tens of thousands of stress granules, we find that the measured fluctuation spectra require an additional contribution, which we attribute to elastic bending deformation. We also show that stress granules have an irregular, nonspherical base shape. These results suggest that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Furthermore, we observe that the measured interfacial tensions and bending rigidities span a range of several orders of magnitude. Hence, different types of stress granules (and more generally, other biomolecular condensates) can only be differentiated via large-scale surveys.

Single active particle in a harmonic potential: Question about the existence of the Jarzynski relation

The interest in active matter stimulates the need to generalize thermodynamic description and relations to active matter systems, which are intrinsically out of equilibrium. One important example is the Jarzynski relation, which links the exponential average of work done in an arbitrary process connecting two equilibrium states with the difference of the free energies of these states. Using a simple model system, a single thermal active Ornstein-Uhlenbeck particle in a harmonic potential, we show that if the standard stochastic thermodynamics definition of work is used, the Jarzynski relation is not generally valid for processes connecting stationary states of active matter systems.

Field-driven cluster formation in two-dimensional colloidal binary mixtures

We study size- and charge-asymmetric oppositely charged colloids driven by an external electric field. The large particles are connected by harmonic springs, forming a hexagonal-lattice network, while the small particles are free of bonds and exhibit fluidlike motion. We show that this model exhibits a cluster formation pattern when the external driving force exceeds a critical value. The clustering is accompanied with stable wave packets in vibrational motions of the large particles.

Boundary design regulates the diffusion of active matter in heterogeneous environments

Physical boundaries play a key role in governing the overall transport properties of nearby self-propelled particles. In this work, we develop dispersion theories and conduct Brownian dynamics simulations to predict the coupling between surface accumulation and effective diffusivity of active particles in boundary-rich media. We focus on three models that are well-understood for passive systems: particle transport in (i) an array of fixed volume-excluding obstacles; (ii) a pore with spatially heterogeneous width; and (iii) a tortuous path with kinks and corners. While the impact of these entropic barriers on passive particle transport is well established, we find that these classical models of porous media flows break down due to the unique interplay between activity and the microstructure of the internal geometry. We study the activity-induced slowdown of effective diffusivity by formulating a Smoluchowski description of long-time self diffusivity which contains contributions from the density and fluctuation fields of the active particles. Particle-based and finite element simulations corroborate this perspective and reveal important nonequilibrium considerations of active transport.

Tuning nonequilibrium phase transitions with inertia

In striking contrast to equilibrium systems, inertia can profoundly alter the structure of active systems. Here, we demonstrate that driven systems can exhibit effective equilibrium-like states with increasing particle inertia, despite rigorously violating the fluctuation-dissipation theorem. Increasing inertia progressively eliminates motility-induced phase separation and restores equilibrium crystallization for active Brownian spheres. This effect appears to be general for a wide class of active systems, including those driven by deterministic time-dependent external fields, whose nonequilibrium patterns ultimately disappear with increasing inertia. The path to this effective equilibrium limit can be complex, with finite inertia sometimes acting to accentuate nonequilibrium transitions. The restoration of near equilibrium statistics can be understood through the conversion of active momentum sources to passive-like stresses. Unlike truly equilibrium systems, the effective temperature is now density dependent, the only remnant of the nonequilibrium dynamics. This density-dependent temperature can in principle introduce departures from equilibrium expectations, particularly in response to strong gradients. Our results provide additional insight into the effective temperature ansatz while revealing a mechanism to tune nonequilibrium phase transitions.

Configurational temperature in active matter. I. Lines of invariant physics in the phase diagram of the Ornstein-Uhlenbeck model

This paper shows that the configurational temperature of liquid-state theory, T_{conf}, defines an energy scale, which can be used for adjusting model parameters of active Ornstein-Uhlenbeck particle (AOUP) models in order to achieve approximately invariant structure and dynamics upon a density change. The required parameter changes are calculated from the variation of a single configuration's T_{conf} for a uniform scaling of all particle coordinates. The resulting equations are justified theoretically for models involving a potential-energy function with hidden scale invariance. The validity of the procedure is illustrated by computer simulations of the Kob-Andersen binary Lennard-Jones AOUP model, showing the existence of lines of approximate invariance of the reduced-unit radial distribution function and time-dependent mean-square displacement.

Harvesting information to control nonequilibrium states of active matter

We propose to use a correlated noise bath to drive an optically trapped Brownian particle that mimics active biological matter. Due to the flexibility and precision of our setup, we are able to control the different parameters that drive the stochastic motion of the particle with unprecedented accuracy, thus reaching strongly correlated regimes that are not easily accessible with real active matter. In particular, by using the correlation time (i.e., the "color") of the noise as a control parameter, we can trigger transitions between two nonequilibrium steady states with no expended work, but only a calorific cost. Remarkably, the measured heat production is directly proportional to the spectral entropy of the correlated noise, in a fashion that is reminiscent of Landauer's principle. Our procedure can be viewed as a method for harvesting information from the active fluctuations.

Active Refrigerators Powered by Inertia

We present the operational principle for a refrigerator that uses inertial effects in active Brownian particles to locally reduce their (kinetic) temperature by 2 orders of magnitude below the environmental temperature. This principle exploits the peculiar but so-far unknown shape of the phase diagram of inertial active Brownian particles to initiate motility-induced phase separation in the targeted cooling regime only. Remarkably, active refrigerators operate without requiring isolating walls opening the route toward using them to systematically absorb and trap, e.g., toxic substances from the environment.

Passive probe particle in an active bath: can we tell it is out of equilibrium?

We study a passive probe immersed in a fluid of active particles. Despite the system's non-equilibrium nature, the trajectory of the probe does not exhibit non-equilibrium signatures: its velocity distribution remains Gaussian, the second fluctuation dissipation theorem is not fundamentally violated, and the motion does not indicate breaking of time reversal symmetry. To tell that the probe is out of equilibrium requires examination of its behavior in tandem with that of the active fluid: the kinetic temperature of the probe does not equilibrate to that of the surrounding active particles. As a strategy to diagnose non-equilibrium from probe trajectories alone, we propose to examine their response to a small perturbation which reveals a non-equilibrium signature through a violation of the first fluctuation dissipation theorem.

Signatures of irreversibility in microscopic models of flocking

Flocking in d=2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

Anomalous Transport of Tracers in Active Baths

We derive the long-time dynamics of a tracer immersed in a one-dimensional active bath. In contrast to previous studies, we find that the damping and noise correlations possess long-time tails with exponents that depend on the tracer symmetry. For generic tracers, shape asymmetry induces ratchet effects that alter fluctuations and lead to superdiffusion and friction that grows with time when the tracer is dragged at a constant speed. In the singular limit of a completely symmetric tracer, we recover normal diffusion and finite friction. Furthermore, for small symmetric tracers, the active contribution to the friction becomes negative: active particles enhance motion rather than oppose it. These results show that, in low-dimensional systems, the motion of a passive tracer in an active bath cannot be modeled as a persistent random walker with a finite correlation time.

Differences in cell death and division rules can alter tissue rigidity and fluidization

Tissue mechanical properties such as rigidity and fluidity, and changes in these properties driven by jamming-unjamming transitions (UJT), have come under recent highlight as mechanical markers of health and disease in various biological processes including cancer. However, most analyses of these mechanical properties and UJT have sidestepped the effect of cellular death and division in these systems. Cellular apoptosis (programmed cell death) and mitosis (cell division) can drive significant changes in tissue properties. The balance between the two is crucial in maintaining tissue function, and an imbalance between the two is seen in situations such as cancer progression, wound healing and necrosis. In this work we investigate the impact of cell death and division on tissue mechanical properties, by incorporating specific mechanosensitive triggers of cell death and division based on the size and geometry of the cell within in silico models of tissue dynamics. Specifically, we look at cell migration, tissue response to external stress, tissue extrusion propensity and self-organization of different cell types within the tissue, as a function of cell death and division and the rules that trigger these events. We find that not only do cell death and division events significantly alter tissue mechanics when compared to systems without these events, but that the choice of triggers driving these cell death and division events also alters the predicted tissue mechanics and overall system behavior.

Analytical approach to chiral active systems: suppressed phase separation of interacting Brownian circle swimmers

We consider chirality in active systems by exemplarily studying the phase behavior of planar systems of interacting Brownian circle swimmers with a spherical shape. For this purpose, we derive a predictive field theory that is able to describe the collective dynamics of circle swimmers. The theory yields a mapping between circle swimmers and noncircling active Brownian particles and predicts that the angular propulsion of the particles leads to a suppression of their motility-induced phase separation, being in line with recent simulation results. In addition, the theory provides analytical expressions for the spinodal corresponding to the onset of motility-induced phase separation and the associated critical point as well as for their dependence on the angular propulsion of the circle swimmers. We confirm our findings by Brownian dynamics simulations. The agreement between results from theory and simulations is found to be good.

Computational exploration of treadmilling and protrusion growth observed in fire ant rafts

Collective living systems regularly achieve cooperative emergent functions that individual organisms could not accomplish alone. The rafts of fire ants (Solenopsis invicta) are often studied in this context for their ability to create aggregated structures comprised entirely of their own bodies, including tether-like protrusions that facilitate exploration of and escape from flooded environments. While similar protrusions are observed in cytoskeletons and cellular aggregates, they are generally dependent on morphogens or external gradients leaving the isolated role of local interactions poorly understood. Here we demonstrate through an ant-inspired, agent-based numerical model how protrusions in ant rafts may emerge spontaneously due to local interactions. The model is comprised of a condensed structural network of agents that represents the monolayer of interconnected worker ants, which floats on the water and gives ant rafts their form. Experimentally, this layer perpetually contracts, which we capture through the pairwise contraction of all neighboring structural agents at a strain rate of [Formula: see text]. On top of the structural layer, we model a dispersed, on-lattice layer of motile agents that represents free ants, which walk on top of the floating network. Experimentally, these self-propelled free ants walk with some mean persistence length and speed that we capture through an ant-inspired phenomenological model. Local interactions occur between neighboring free ants within some radius of detection, R, and the persistence length of freely active agents is tuned through a noise parameter, η as introduced by the Vicsek model. Both R and η where fixed to match the experimental trajectories of free ants. Treadmilling of the raft occurs as agents transition between the structural and free layers in accordance with experimental observations. Ultimately, we demonstrate how phases of exploratory protrusion growth may be induced by increased ant activity as characterized by a dimensionless parameter, [Formula: see text]. These results provide an example in which functional morphogenesis of a living system may emerge purely from local interactions at the constituent length scale, thereby providing a source of inspiration for the development of decentralized, autonomous active matter and swarm robotics.

Ising-like Critical Behavior of Vortex Lattices in an Active Fluid

Turbulent vortex structures emerging in bacterial active fluids can be organized into regular vortex lattices by weak geometrical constraints such as obstacles. Here we show, using a continuum-theoretical approach, that the formation and destruction of these patterns exhibit features of a continuous second-order equilibrium phase transition, including long-range correlations, divergent susceptibility, and critical slowing down. The emerging vorticity field can be mapped onto a two-dimensional (2D) Ising model with antiferromagnetic nearest-neighbor interactions by coarse graining. The resulting effective temperature is found to be proportional to the strength of the nonlinear advection in the continuum model.

Tracer dynamics in crowded active-particle suspensions

We discuss the dynamics of active Brownian particles (ABPs) in crowded environments through the mean-squared displacement (MSD) of active and passive tracer particles in both active and passive host systems. Exact equations for the MSD are derived using a projection operator technique, extending to dense systems the known solution for a single ABP. The interaction of the tracer particle with the host particles gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for active Brownian particles (ABP-MCT), we discuss the various dynamical regimes that emerge: While self-propelled motion gives rise to super-diffusive MSD, at high densities, this competes with an interaction-induced sub-diffusive regime. The predictions of the theory are shown to be in good agreement with results obtained from an event-driven Brownian dynamics (ED-BD) simulation scheme for the dynamics of two-dimensional active Brownian hard disks.

Dynamic and mechanical evolution of an oil-water interface during bacterial biofilm formation

We present an experimental study combining particle tracking, active microrheology, and differential dynamic microscopy (DDM) to investigate the dynamics and rheology of an oil-water interface during biofilm formation by the bacteria Pseudomonas Aeruginosa PA14. The interface transitions from an active fluid dominated by the swimming motion of adsorbed bacteria at early age to an active viscoelastic system at late ages when the biofilm is established. The microrheology measurements using microscale magnetic rods indicate that the biofilm behaves as a viscoelastic solid at late age. The bacteria motility at the interface during the biofilm formation, which is characterized in the DDM measurements, evolves from diffusive motion at early age to constrained, quasi-localized motion at later age. Similarly, the mobility of passively moving colloidal spheres at the interface decreases significantly with increasing interface age and shows a dependence on sphere size after biofilm formation that is orders-of-magnitude larger than that expected in a homogeneous system in equilibrium. We attribute this anomalous size dependence to either length-scale-dependent rheology of the biofilm or widely differing effects of the bacteria activity on the motion of spheres of different sizes.

Emergence of effective temperatures in an out-of-equilibrium model of biopolymer folding

We investigate the possibility of extending the notion of temperature in a stochastic model for the RNA or protein folding driven out of equilibrium. We simulate the dynamics of a small RNA hairpin subject to an external pulling force, which is time-dependent. First, we consider a fluctuation-dissipation relation (FDR) whereby we verify that various effective temperatures can be obtained for different observables, only when the slowest intrinsic relaxation timescale of the system regulates the dynamics of the system. Then, we introduce a different nonequilibrium temperature, which is defined from the rate of heat exchanged with a weakly interacting thermal bath. Notably, this "kinetic" temperature can be defined for any frequency of the external switching force. We also discuss and compare the behavior of these two emerging parameters, by discriminating the time-delayed nature of the FDR temperature from the instantaneous character of the kinetic temperature. The validity of our numerics are corroborated by a simple four-state Markov model which describes the long-time behavior of the RNA molecule.

Unifying fluctuation-dissipation temperatures of slow-evolving nonequilibrium systems from the perspective of inherent structures

For nonequilibrium systems, how to define temperature is one of the key and difficult issues to solve. Although effective temperatures have been proposed and studied to this end, it still remains elusive what they actually are. Here, we focus on the fluctuation-dissipation temperatures and report that such effective temperatures of slow-evolving systems represent characteristic temperatures of their equilibrium counterparts. By calculating the fluctuation-dissipation relation of inherent structures, we obtain a temperature-like quantity T IS For monocomponent crystal-formers, T IS agrees well with the crystallization temperature T c, while it matches with the onset temperature T on for glass-formers. It also agrees with effective temperatures of typical nonequilibrium systems, such as aging glasses, quasi-static shear flows, and quasi-static self-propelled flows. From the unique perspective of inherent structures, our study reveals the nature of effective temperatures and the underlying connections between nonequilibrium and equilibrium systems and confirms the equivalence between T on and T c.

The Einstein effective temperature can predict the tagged active particle density

We derive a distribution function for the position of a tagged active particle in a slowly varying in space external potential, in a system of interacting active particles. The tagged particle distribution has the form of the Boltzmann distribution but with an effective temperature that replaces the temperature of the heat bath. We show that the effective temperature that enters the tagged particle distribution is the same as the effective temperature defined through the Einstein relation, i.e., it is equal to the ratio of the self-diffusion and tagged particle mobility coefficients. This result shows that this effective temperature, which is defined through a fluctuation-dissipation ratio, is relevant beyond the linear response regime. We verify our theoretical findings through computer simulations. Our theory fails when an additional large length scale appears in our active system. In the system we simulated, this length scale is associated with long-wavelength density fluctuations that emerge upon approaching motility-induced phase separation.

How do clusters in phase-separating active matter systems grow? A study for Vicsek activity in systems undergoing vapor-solid transition

Via molecular dynamics simulations, we have studied the kinetics of vapor-"solid" phase transition in an active matter model in which self-propulsion is introduced via the well-known Vicsek rule. The overall density of the particles is chosen in such a way that the evolution morphology consists of disconnected clusters that are defined as regions of high density of particles. Our focus has been on understanding the influence of the above-mentioned self-propulsion on structure and growth of these clusters by comparing the results with those for the passive limit of the model that also exhibits vapor-"solid" transition. While in the passive case growth occurs due to a standard diffusive mechanism, the Vicsek activity leads to very rapid growth, via a process that is practically equivalent to the ballistic aggregation mechanism. The emerging growth law in the latter case has been accurately estimated and explained by invoking information on velocity and structural aspects of the clusters into a relevant theory. Some of these results are also discussed with reference to a model for active Brownian particles.

Lamellar to Micellar Phases and Beyond: When Tactic Active Systems Admit Free Energy Functionals

We consider microscopic models of active particles whose velocities, rotational diffusivities, and tumbling rates depend on the gradient of a local field that is either externally imposed or depends on all particle positions. Despite the fundamental differences between active and passive dynamics at the microscopic scale, we show that a large class of such tactic active systems admit fluctuating hydrodynamics equivalent to those of interacting Brownian colloids in equilibrium. We exploit this mapping to show how taxis may lead to the lamellar and micellar phases observed for soft repulsive colloids. In the context of chemotaxis, we show how the competition between chemoattractant and chemorepellent may lead to a bona fide equilibrium liquid-gas phase separation in which a loss of thermodynamic stability of the fluid signals the onset of a chemotactic collapse.

Anomalous diffusion of active Brownian particles cross-linked to a networked polymer: Langevin dynamics simulation and theory

Quantitatively understanding the dynamics of an active Brownian particle (ABP) interacting with a viscoelastic polymer environment is a scientific challenge. It is intimately related to several interdisciplinary topics such as the microrheology of active colloids in a polymer matrix and the athermal dynamics of the in vivo chromosomes or cytoskeletal networks. Based on Langevin dynamics simulation and analytic theory, here we explore such a viscoelastic active system in depth using a star polymer of functionality f with the center cross-linker particle being ABP. We observe that the ABP cross-linker, despite its self-propelled movement, attains an active subdiffusion with the scaling ΔR2(t) ∼ tα with α ≤ 1/2, through the viscoelastic feedback from the polymer. Counter-intuitively, the apparent anomaly exponent α becomes smaller as the ABP is driven by a larger propulsion velocity, but is independent of functionality f or the boundary conditions of the polymer. We set forth an exact theory and show that the motion of the active cross-linker is a Gaussian non-Markovian process characterized by two distinct power-law displacement correlations. At a moderate Péclet number, it seemingly behaves as fractional Brownian motion with a Hurst exponent H = α/2, whereas, at a high Péclet number, the self-propelled noise in the polymer environment leads to a logarithmic growth of the mean squared displacement (∼ln t) and a velocity autocorrelation decaying as -t-2. We demonstrate that the anomalous diffusion of the active cross-linker is precisely described by a fractional Langevin equation with two distinct random noises.

Isomorph theory beyond thermal equilibrium

This paper generalizes isomorph theory to systems that are not in thermal equilibrium. The systems are assumed to be R-simple, i.e., to have a potential energy that as a function of all particle coordinates R obeys the hidden-scale-invariance condition U(R_a) < U(R_b) ⇒ U(λR_a) < U(λR_b). "Systemic isomorphs" are introduced as lines of constant excess entropy in the phase diagram defined by density and systemic temperature, which is the temperature of the equilibrium state point with the average potential energy equal to U(R). The dynamics is invariant along a systemic isomorph if there is a constant ratio between the systemic and the bath temperature. In thermal equilibrium, the systemic temperature is equal to the bath temperature and the original isomorph formalism is recovered. The new approach rationalizes within a consistent framework previously published observations of isomorph invariance in simulations involving nonlinear steady-state shear flows, zero-temperature plastic flows, and glass-state isomorphs. This paper relates briefly to granular media, physical aging, and active matter. Finally, we discuss the possibility that the energy unit defining the reduced quantities should be based on the systemic rather than the bath temperature.

Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems

We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single-particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent of the friction coefficient.

Coordinate transformation methodology for simulating quasistatic elastoplastic solids

Molecular dynamics simulations frequently employ periodic boundary conditions where the positions of the periodic images are manipulated in order to apply deformation to the material sample. For example, Lees-Edwards conditions use moving periodic images to apply simple shear. Here, we examine the problem of precisely comparing this type of simulation to continuum solid mechanics. We employ a hypoelastoplastic mechanical model, and develop a projection method to enforce quasistatic equilibrium. We introduce a simulation framework that uses a fixed Cartesian computational grid on a reference domain, and which imposes deformation via a time-dependent coordinate transformation to the physical domain. As a test case for our method, we consider the evolution of shear bands in a bulk metallic glass using the shear transformation zone theory of amorphous plasticity. We examine the growth of shear bands in simple shear and pure shear conditions as a function of the initial preparation of the bulk metallic glass.

Active noise experienced by a passive particle trapped in an active bath

We study the properties of active noise experienced by a passive particle harmonically trapped in an active bath. The active bath is either explicitly simulated by an ensemble of active Brownian particles or abstractly represented by an active colored noise in theory. Assuming the equivalence of the two descriptions of the active bath, the active noise in the simulation system, which is directly extracted by fitting theoretical predictions to simulation measurements, is shown to depend on the constraint suffered by the passive tracer. This scenario is in significant contrast to the case of thermal noise that is independent of external trap potentials. The constraint dependence of active noise arises from the fact that the persistent force on the passive particle from the active bath can be influenced by the particle relaxation dynamics. Moreover, due to the interplay between the active collisions and particle relaxation dynamics, the effective temperature of the passive tracer quantified as the ratio of fluctuation to dissipation increases as the constraint strengthens, while the average potential and kinetic energies of the passive particle both decrease.

A practical guide to active colloids: choosing synthetic model systems for soft matter physics research

Synthetic active colloids that harvest energy stored in the environment and swim autonomously are a popular model system for active matter. This emerging field of research sits at the intersection of materials chemistry, soft matter physics, and engineering, and thus cross-talk among researchers from different backgrounds becomes critical yet difficult. To facilitate this interdisciplinary communication, and to help soft matter physicists with choosing the best model system for their research, we here present a tutorial review article that describes, in appropriate detail, six experimental systems of active colloids commonly found in the physics literature. For each type, we introduce their background, material synthesis and operating mechanisms and notable studies from the soft matter community, and comment on their respective advantages and limitations. In addition, the main features of each type of active colloid are summarized into two useful tables. As materials chemists and engineers, we intend for this article to serve as a practical guide, so those who are not familiar with the experimental aspects of active colloids can make more informed decisions and maximize their creativity.

Constraint Dependence of Active Depletion Forces on Passive Particles

Using simulations and experiments, we demonstrate that the effective interaction between passive particles in an active bath substantially depends on an external constraint suffered by the passive particles. Particularly, the effective interaction between two free passive particles, which is directly measured in simulation, is qualitatively different from the one between two fixed particles. Moreover, we find that the friction experienced by the passive particles-a kinematic constraint-similarly influences the effective interaction. These remarkable features are in significant contrast to the equilibrium cases, and mainly arise from the accumulation of the active particles near the concave gap formed by the passive spheres. This constraint dependence not only deepens our understanding of the "active depletion force," but also provides an additional tool to tune the effective interactions in an active bath.

Phase transition and emergence of active temperature in an active Brownian system in underdamped background

We explore the role of inertia in the properties of active Brownian particles (ABPs) immersed in an underdamped background in two dimensions using Langevin dynamics computer simulation. Similar to an equilibrium two-dimensional passive interacting particle system, the system of ABPs transits from a liquid phase to a solid phase with the change in the coupling parameter, which is the ratio of interaction potential energy and thermal energy of the background solvent. Important qualitative and quantitative differences are found in the liquid-solid phase transition with increasing strength of activity as compared to those found in the conventional overdamped background limit. In the underdamped background, inherent activity is found to lead to a temperature, called the active temperature and defined by average velocity fluctuations of the ABPs, that is different from the fixed background solvent temperature. A new scaling law for active temperature as a function of activity strength is found near the liquid-solid boundary. Active temperature, which behaves similar to the thermodynamic equilibrium temperature, is also found to depend upon the interaction strength between the active particles and the strength of the background dissipation. With an increase in background dissipation, the difference between active temperature and the background solvent temperature decreases and the difference is found to eventually vanish in the overdamped limit, demonstrating the correctness of the calculation.

Activated diffusiophoresis

Perturbations of fluid media can give rise to non-equilibrium dynamics, which may, in turn, cause motion of immersed inclusions or tracer particles. We consider perturbations ("activations") that are local in space and time, of a fluid density which is conserved, and study the resulting diffusiophoretic phenomena that emerge at a large distance. Specifically, we consider cases where the perturbations propagate diffusively, providing examples from passive and active matter for which this is expected to be the case. Activations can, for instance, be realized by sudden and local changes in interaction potentials of the medium or by local changes in its activity. Various analytical results are provided for the case of confinement by two parallel walls. We investigate the possibility of extracting work from inclusions, which are moving through the activated fluid. Furthermore, we show that a time-dependent density profile, created via suitable activation protocols, allows for the conveyance of inclusions along controlled and stable trajectories. In contrast, in states with a steady density, inclusions cannot be held at stable positions, reminiscent of Earnshaw's theorem of electrostatics. We expect these findings to be applicable in a range of experimental systems. The phenomena described here are argued to be distinct from other forms of phoresis such as thermophoresis.

Active binary mixtures of fast and slow hard spheres

We computationally studied the phase behavior and dynamics of binary mixtures of active particles, where each species had distinct activities leading to distinct velocities, fast and slow. We obtained phase diagrams demonstrating motility-induced phase separation (MIPS) upon varying the activity and concentration of each species, and extended current kinetic theory of active/passive mixtures to active/active mixtures. We discovered two regimes of behavior quantified through the participation of each species in the dense phase compared to their monodisperse counterparts. In regime I (active/passive and active/weakly-active), we found that the dense phase was segregated by particle type into domains of fast and slow particles. Moreover, fast particles were suppressed from entering the dense phase while slow particles were enhanced entering the dense phase, compared to monodisperse systems of all-fast or all-slow particles. These effects decayed asymptotically as the activity of the slow species increased, approaching the activity of the fast species until they were negligible (regime II). In regime II, the dense phase was homogeneously mixed and each species participated in the dense phase as if it were it a monodisperse system (i.e. not mixed at all). Finally, we showed that a weighted average of constituent particle activities, which we term the net activity, defines a binodal for the MIPS transition in active/active binary mixtures. We examined the critical point of the transition and found a critical exponent (β = 0.45) in agreement with similar studies on monodisperse systems, and distinct from equilibrium systems.

Linear Response Theory and Green-Kubo Relations for Active Matter

We address the question of how interacting active systems in a nonequilibrium steady state respond to an external perturbation. We establish an extended fluctuation-dissipation theorem for active Brownian particles (ABP), which highlights the role played by the local violation of detailed balance due to activity. By making use of a Markovian approximation we derive closed Green-Kubo expressions for the diffusivity and mobility of ABP and quantify the deviations from the Stokes-Einstein relation. We compute the linear response function to an external force using unperturbed simulations of ABP and compare the results with the analytical predictions of the transport coefficients. Our results show the importance of the interplay between activity and interactions in the departure from equilibrium linear response.

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. However, the previous TUR for underdamped dynamics is not applicable to a system driven by a velocity-dependent force. In this study, we present a TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [Phys. Rev. E 100, 032130 (2019)2470-004510.1103/PhysRevE.100.032130]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Directional Stepping Model for Yeast Dynein: Longitudinal- and Side-Step Distributions

Motor proteins are biological machines that convert chemical energy stored in ATP to mechanical work. Kinesin and dynein are microtubule (MT)-associated motor proteins that, among other functions, facilitate intracellular transport. Here, we focus on dynein motility. We deduce the directional step distribution of yeast dynein motor protein on the MT surface by combing intrinsic features of the dynein and MTs. These include the probability distribution of the separation vector between the two microtubule-binding domains, the angular probability distribution of a single microtubule-binding domain translation, the existence of an MT seam defect, MT-binding sites, and theoretical extension that accounts for a load force on the motor. Our predictions are in excellent accord with the measured longitudinal step size distributions at various load forces. Moreover, we predict the side-step distribution and its dependence on longitudinal load forces, which shows a few surprising features. First, the distribution is broad. Second, in the absence of load, we find a small right-handed bias. Third, the side-step bias is susceptible to the longitudinal load force; it vanishes at a load equal to the motor stalling force and changes to a left-hand bias above that value. Fourth, our results are sensitive to the ability of the motor to explore the seam several times during its walk. Although available measurements of side-way distribution are limited, our findings are amenable to experimental check and, moreover, suggest a diversity of results depending on whether the MT seam is viable to motor sampling.

Generalized Ornstein-Uhlenbeck model for active motion

We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E 90, 012111 (2014)10.1103/PhysRevE.90.012111]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.