Multidimensional stationary probability distribution for interacting active particles

Noise-Induced Phase Separation and Time Reversal Symmetry Breaking in Active Field Theories Driven by Persistent Noise

Within the Landau-Ginzburg picture of phase transitions, scalar field theories develop phase separation because of a spontaneous symmetry-breaking mechanism. This picture works in thermodynamics but also in the dynamics of phase separation. Here we show that scalar nonequilibrium field theories undergo phase separation just because of nonequilibrium fluctuations driven by a persistent noise. The mechanism is similar to what happens in motility-induced phase separation where persistent motion introduces an effective attractive force. We observe that noise-induced phase separation occurs in a region of the phase diagram where disordered field configurations would otherwise be stable at equilibrium. Measuring the local entropy production rate to quantify the time-reversal symmetry breaking, we find that such breaking is concentrated on the boundary between the two phases.

Multiple temperatures and melting of a colloidal active crystal

Thermal fluctuations constantly excite all relaxation modes in an equilibrium crystal. As the temperature rises, these fluctuations promote the formation of defects and eventually melting. In active solids, the self-propulsion of "atomic" units provides an additional source of non-equilibrium fluctuations whose effect on the melting scenario is still largely unexplored. Here we show that when a colloidal crystal is activated by a bath of swimming bacteria, solvent temperature and active temperature cooperate to define dynamic and thermodynamic properties. Our system consists of repulsive paramagnetic particles confined in two dimensions and immersed in a bath of light-driven E. coli. The relative balance between fluctuations and interactions can be adjusted in two ways: by changing the strength of the magnetic field and by tuning activity with light. When the persistence time of active fluctuations is short, a single effective temperature controls both the amplitudes of relaxation modes and the melting transition. For more persistent active noise, energy equipartition is broken and multiple temperatures emerge, whereas melting occurs before the Lindemann parameter reaches its equilibrium critical value. We show that this phenomenology is fully confirmed by numerical simulations and framed within a minimal model of a single active particle in a periodic potential.

Recovering the activity parameters of an active fluid confined in a sphere

The properties of an active fluid, for example, a bacterial bath or a collection of microtubules and molecular motors, can be accessed through the dynamics of passive particle probes. Here, in the perspective of analyzing experimental situations of confinement in droplets, we consider the kinematics of a negatively buoyant probe particle in an active fluid, both confined within a spherical domain. The active bath generates a fluctuating flow that pushes the particle with a velocity that is modeled as a colored stochastic noise, characterized by two parameters, the intensity and memory time of the active flow. When the particle departs a little from the bottom of the spherical domain, the configuration is well approximated by a particle in a two-dimensional harmonic trap subjected to the colored noise, in which case an analytical solution exists, which is the base for quantitative analysis. We numerically simulate the dynamics of the particle and use the planar, two-dimensional mean square displacement to recover the activity parameters of the bath. This approach yields satisfactory results as long as the particle remains relatively confined; that is, as long as the intensity of the colored noise remains low.

Active oscillatory associative memory

Traditionally, physical models of associative memory assume conditions of equilibrium. Here, we consider a prototypical oscillator model of associative memory and study how active noise sources that drive the system out of equilibrium, as well as nonlinearities in the interactions between the oscillators, affect the associative memory properties of the system. Our simulations show that pattern retrieval under active noise is more robust to the number of learned patterns and noise intensity than under passive noise. To understand this phenomenon, we analytically derive an effective energy correction due to the temporal correlations of active noise in the limit of short correlation decay time. We find that active noise deepens the energy wells corresponding to the patterns by strengthening the oscillator couplings, where the more nonlinear interactions are preferentially enhanced. Using replica theory, we demonstrate qualitative agreement between this effective picture and the retrieval simulations. Our work suggests that the nonlinearity in the oscillator couplings can improve memory under nonequilibrium conditions.

Time-dependent properties of run-and-tumble particles: Density relaxation

We characterize collective diffusion of hardcore run-and-tumble particles (RTPs) by explicitly calculating the bulk-diffusion coefficient D(ρ,γ) for arbitrary density ρ and tumbling rate γ, in systems on a d-dimensional periodic lattice. We study two minimal models of RTPs: Model I is the standard version of hardcore RTPs introduced in [Phys. Rev. E 89, 012706 (2014)10.1103/PhysRevE.89.012706], whereas model II is a long-ranged lattice gas (LLG) with hardcore exclusion, an analytically tractable variant of model I. We calculate the bulk-diffusion coefficient analytically for model II and numerically for model I through an efficient Monte Carlo algorithm; notably, both models have qualitatively similar features. In the strong-persistence limit γ→0 (i.e., dimensionless ratio r_{0}γ/v→0), with v and r_{0} being the self-propulsion speed and particle diameter, respectively, the fascinating interplay between persistence and interaction is quantified in terms of two length scales: (i) persistence length l_{p}=v/γ and (ii) a "mean free path," being a measure of the average empty stretch or gap size in the hopping direction. We find that the bulk-diffusion coefficient varies as a power law in a wide range of density: D∝ρ^{-α}, with exponent α gradually crossing over from α=2 at high densities to α=0 at low densities. As a result, the density relaxation is governed by a nonlinear diffusion equation with anomalous spatiotemporal scaling. In the thermodynamic limit, we show that the bulk-diffusion coefficient-for ρ,γ→0 with ρ/γ fixed-has a scaling form D(ρ,γ)=D^{(0)}F(ρav/γ), where a∼r_{0}^{d-1} is particle cross section and D^{(0)} is proportional to the diffusion coefficient of noninteracting particles; the scaling function F(ψ) is calculated analytically for model II (LLG) and numerically for model I. Our arguments are independent of dimensions and microscopic details.

Anisotropic run-and-tumble-turn dynamics

Run-and-tumble processes successfully model several living systems. While studies have typically focused on particles with isotropic tumbles, recent examples exhibit "tumble-turns", in which particles undergo 90° tumbles and so possess explicitly anisotropic dynamics. We study the consequences of such tumble-turn anisotropicity at both short and long-time scales. We model run-and-tumble-turn particles as self-propelled particles subjected to an angular potential that favors directions of movement parallel to Cartesian axes. Using agent-based simulations, we study the effects of the interplay between rotational diffusion and an aligning potential on the particles' trajectories, which leads to the right-angled turns. We demonstrate that the long-time effect is to alter the tumble-turn time, which governs the long-time dynamics. In particular, when normalized by this timescale, trajectories become independent of the underlying details of the potential. As such, we develop a simplified continuum theory, which quantitatively agrees with agent-based simulations. We find that the purely diffusive hydrodynamic limit still exhibits anisotropic features at intermediate times and conclude that the transition to diffusive dynamics precedes the transition to isotropic dynamics. By considering short-range repulsive and alignment particle-particle interactions, we show how the anisotropic features of a single particle are inherited by the global order of the system. We hope this work will shed light on how active systems can extend local anisotropic properties to macroscopic scales, which might be important in biological processes occurring in anisotropic environments.

Universal Poisson statistics of a passive tracer diffusing in dilute active suspensions

The statistics of a passive tracer immersed in a suspension of active particles (swimmers) is derived from first principles by considering a perturbative expansion of the tracer interaction with the microscopic swimmer field. To first order in the swimmer density, the tracer statistics is shown to be exactly represented by a spatial Poisson process combined with independent tracer-swimmer scattering events, rigorously reducing the multiparticle dynamics to two-body interactions. The Poisson representation is valid in any dimension, for arbitrary interaction forces and for a large class of swimmer dynamics. The framework not only allows for the systematic calculation of the tracer statistics in various dynamical regimes but highlights in particular surprising universal features that are independent of the swimmer dynamics such as a time-independent velocity distribution.

Work Fluctuations for a Harmonically Confined Active Ornstein-Uhlenbeck Particle

We study the active work fluctuations of an active Ornstein-Uhlenbeck particle in the presence of a confining harmonic potential. We tackle the problem analytically both for stationary and generic uncorrelated initial states. Our results show that harmonic confinement can induce singularities in the active work rate function, with linear stretches at large positive and negative active work, at sufficiently large active and harmonic force constants. These singularities originate from big jumps in the displacement and in the active force, occurring at the initial or ending points of trajectories and marking the relevance of boundary terms in this problem.

Control of Active Brownian Particles: An Exact Solution

Control of stochastic systems is a challenging open problem in statistical physics, with a wealth of potential applications from biology to granulates. Unlike most cases investigated so far, we aim here at controlling a genuinely out-of-equilibrium system, the two dimensional active Brownian particles model in a harmonic potential, a paradigm for the study of self-propelled bacteria. We search for protocols for the driving parameters (stiffness of the potential and activity of the particles) bringing the system from an initial passivelike steady state to a final activelike one, within a chosen time interval. The exact analytical results found for this prototypical model of self-propelled particles brings control techniques to a wider class of out-of-equilibrium systems.

Chiral active matter in external potentials

We investigate the interplay between chirality and confinement induced by the presence of an external potential. For potentials having radial symmetry, the circular character of the trajectories induced by the chiral motion reduces the spatial fluctuations of the particle, thus providing an extra effective confining mechanism, that can be interpreted as a lowering of the effective temperature. In the case of non-radial potentials, for instance, with an elliptic shape, chirality displays a richer scenario. Indeed, the chirality can break the parity symmetry of the potential that is always fulfilled in the non-chiral system. The probability distribution displays a strong non-Maxwell-Boltzmann shape that emerges in cross-correlations between the two Cartesian components of the position, that vanishes in the absence of chirality or when radial symmetry of the potential is restored. These results are obtained by considering two popular models in active matter, i.e. chiral Active Brownian particles and chiral active Ornstein-Uhlenbeck particles.

Equation of state for active matter

We characterize the steady states of a suspension of two-dimensional active Brownian particles (ABPs). By approximating the first-order correction to the steady-state probability distribution to lowest order in Peclet number, we show that macroscopic quantities can be calculated in analogous way to equilibrium systems using this probability distribution. We then derive expressions for the macroscopic pressure and position-orientation correlation functions. We check our results by direct comparison with extensive numerical simulations. A key finding is the importance of many-body effective interactions even at very low densities.

Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties

We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behavior is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modeling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.

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.

Most probable paths for active Ornstein-Uhlenbeck particles

Fluctuations play an important role in the dynamics of stochastic systems. In particular, for small systems, the most probable thermodynamic quantities differ from their averages because of the fluctuations. Using the Onsager Machlup variational formalism we analyze the most probable paths for nonequilibrium systems, in particular, active Ornstein-Uhlenbeck particles, and investigate how the entropy production along these paths differs from the average entropy production. We investigate how much information about their nonequilibrium nature can be obtained from their extremum paths and how these paths depend on the persistence time and their swim velocities. We also look at how the entropy production along the most probable paths varies with the active noise and how it differs from the average entropy production. This study would be useful to design artificial active systems with certain target trajectories.

Dynamics of active particles with translational and rotational inertia

Inertial effects affecting both the translational and rotational dynamics are inherent to a broad range of active systems at the macroscopic scale. Thus, there is a pivotal need for proper models in the framework of active matter to correctly reproduce experimental results, hopefully achieving theoretical insights. For this purpose, we propose an inertial version of the active Ornstein-Uhlenbeck particle (AOUP) model accounting for particle mass (translational inertia) as well as its moment of inertia (rotational inertia) and derive the full expression for its steady-state properties. The inertial AOUP dynamics introduced in this paper is designed to capture the basic features of the well-established inertial active Brownian particle model, i.e. the persistence time of the active motion and the long-time diffusion coefficient. For a small or moderate rotational inertia, these two models predict similar dynamics at all timescales and, in general, our inertial AOUP model consistently yields the same trend upon changing the moment of inertia for various dynamical correlation functions.

Flocking without Alignment Interactions in Attractive Active Brownian Particles

Within a simple model of attractive active Brownian particles, we predict flocking behavior and challenge the widespread idea that alignment interactions are necessary to observe this collective phenomenon. Here, we show that even nonaligning attractive interactions can lead to a flocking state. Monitoring the velocity polarization as the order parameter, we reveal the onset of a first-order transition from a disordered phase, characterized by several small clusters, to a flocking phase, where a single flocking cluster is emerging. The scenario is confirmed by studying the spatial connected correlation function of particle velocities, which reveals scale-free behavior in flocking states and exponential-like decay for nonflocking configurations. Our predictions can be tested in microscopic and macroscopic experiments showing flocking, such as animals, migrating cells, and active colloids.

Classical Nucleation Theory for Active Fluid Phase Separation

Classical Nucleation Theory (CNT), linking rare nucleation events to the free-energy landscape of a growing nucleus, is central to understanding phase-change kinetics in passive fluids. Nucleation in nonequilibrium systems is much harder to describe because there is no free energy, but instead a dynamics-dependent quasipotential that typically must be found numerically. Here we extend CNT to a class of active phase-separating systems governed by a minimal field-theoretic model (Active Model B+). In the small noise and supersaturation limits that CNT assumes, we compute analytically the quasipotential, and hence, nucleation barrier, for liquid-vapor phase separation. Crucial to our results, detailed balance, although broken microscopically by activity, is restored along the instanton trajectory, which in CNT involves the nuclear radius as the sole reaction coordinate.

Most probable path of active Ornstein-Uhlenbeck particles

Using the path integral representation of the nonequilibrium dynamics, we compute the most probable path between arbitrary starting and final points that is followed by an active particle driven by persistent noise. We focus our attention on the case of active particles immersed in harmonic potentials, where the trajectory can be computed analytically. Once we consider the extended Markovian dynamics where the self-propulsive drive evolves according to an Ornstein-Uhlenbeck process, we can compute the trajectory analytically with arbitrary conditions on position and self-propulsion velocity. We test the analytical predictions against numerical simulations and we compare the analytical results with those obtained within approximated equilibriumlike dynamics.

Glassy Dynamics in Chiral Fluids

Chiral active matter is enjoying a rapid increase of interest, spurred by the rich variety of asymmetries that can be attained in, e.g., the shape or self-propulsion mechanism of active particles. Though this has already led to the observance of so-called chiral crystals, active chiral glasses remain largely unexplored. A possible reason for this could be the naive expectation that interactions dominate the glassy dynamics and the details of the active motion become increasingly less relevant. Here, we show that quite the opposite is true by studying the glassy dynamics of interacting chiral active Brownian particles. We demonstrate that when our chiral fluid is pushed to glassy conditions, it exhibits highly nontrivial dynamics, especially compared to a standard linear active fluid such as common active Brownian particles. Despite the added complexity, we are still able to present a full rationalization for all identified dynamical regimes. Most notably, we introduce a new "hammering" mechanism, unique to rapidly spinning particles in high-density conditions, that can fluidize a chiral active solid.

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.

Bistability in orbital trajectories of a chiral self-propelled particle interacting with an external field

In this paper, the dynamics of a self-propelled stochastic particle under the influence of an axisymmetric light field is experimentally studied. The particle under consideration has the main characteristic of carrying a light sensor in an eccentric location. For the chosen experimental conditions, the emerging trajectories are orbital, and, more interestingly, they suggest the existence of bistability. A mathematical model incorporating the key experimental components is introduced. By means of numerical simulations and theoretical analysis, it is found that, in addition to the orbiting behavior, the sensor location could produce trapped or diffusive behaviors. Furthermore, the study reveals that stochastic perturbation and the eccentric location of the sensor are responsible for inducing bistability in the orbital trajectories, supporting experimental observations.

From predicting to learning dissipation from pair correlations of active liquids

Active systems, which are driven out of equilibrium by local non-conservative forces, can adopt unique behaviors and configurations. Towards designing such materials, an important challenge is to precisely connect the static structure of active systems to the dissipation of energy induced by the local driving. Here, we use tools from liquid-state theories and machine learning to take on these challenges. We first demonstrate analytically for an isotropic active matter system that dissipation and pair correlations are closely related when driving forces behave like an active temperature. We then extend a nonequilibrium mean-field framework for predicting these pair correlations which, unlike most existing approaches, is applicable even for strongly interacting particles and far from equilibrium, to predict dissipation in these systems. Based on this theory, we reveal analytically a robust relation between dissipation and structure which holds even as the system approaches a nonequilibrium phase transition. Finally, we construct a neural network which maps static configurations of particles to their dissipation rate without any prior knowledge of the underlying dynamics. Our results open novel perspectives on the interplay between dissipation and organization out-of-equilibrium.

Mean-field theory for the structure of strongly interacting active liquids

Active systems, which are driven out of equilibrium by local non-conservative forces, exhibit unique behaviors and structures with potential utility for the design of novel materials. An important and difficult challenge along the path towards such a goal is to precisely predict how the structure of active systems is modified as their driving forces push them out of equilibrium. Here, we use tools from liquid-state theories to approach this challenge for a classic minimal isotropic active matter model. First, we construct a nonequilibrium mean-field framework which can predict the structure of systems of weakly interacting particles. Second, motivated by equilibrium solvation theories, we modify this theory to extend it with surprisingly high accuracy to strongly interacting particles, distinguishing it from most existing similarly tractable approaches. Our results provide insight into spatial organization in strongly interacting out-of-equilibrium systems and strategies to control them.

Dynamics of active particles with space-dependent swim velocity

We study the dynamical properties of an active particle subject to a swimming speed explicitly depending on the particle position. The oscillating spatial profile of the swim velocity considered in this paper takes inspiration from experimental studies based on Janus particles whose speed can be modulated by an external source of light. We suggest and apply an appropriate model of an active Ornstein Uhlenbeck particle (AOUP) to the present case. This allows us to predict the stationary properties, by finding the exact solution of the steady-state probability distribution of particle position and velocity. From this, we obtain the spatial density profile and show that its form is consistent with the one found in the framework of other popular models. The reduced velocity distribution highlights the emergence of non-Gaussianity in our generalized AOUP model which becomes more evident as the spatial dependence of the velocity profile becomes more pronounced. Then, we focus on the time-dependent properties of the system. Velocity autocorrelation functions are studied in the steady-state combining numerical and analytical methods derived under suitable approximations. We observe a non-monotonic decay in the temporal shape of the velocity autocorrelation function which depends on the ratio between the persistence length and the spatial period of the swim velocity. In addition, we numerically and analytically study the mean square displacement and the long-time diffusion coefficient. The ballistic regime, observed in the small-time region, is deeply affected by the properties of the swim velocity landscape which induces also a crossover to a sub-ballistic but superdiffusive regime for intermediate times. Finally, the long-time diffusion coefficient decreases as the amplitude of the swim velocity oscillations increases because the diffusion is mainly determined by those regions where the particles are slow.

Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter

Dense active matter is gaining widespread interest due to its remarkable similarity with conventional glass-forming materials. However, active matter is inherently out of equilibrium and even simple models such as active Brownian particles (ABPs) and active Ornstein-Uhlenbeck particles (AOUPs) behave markedly differently from their passive counterparts. Controversially, this difference has been shown to manifest itself via either a speedup, slowdown, or nonmonotonic change of the glassy relaxation dynamics. Here we rationalize these seemingly contrasting views on the departure from equilibrium by identifying the ratio of the short-time length scale to the cage length, i.e., the length scale of local particle caging, as a vital and unifying control parameter for active glassy matter. In particular, we explore the glassy dynamics of both thermal and athermal ABPs and AOUPs upon increasing the persistence time. We find that for all studied systems there is an optimum of the dynamics; this optimum occurs when the cage length coincides with the corresponding short-time length scale of the system, which is either the persistence length for athermal systems or a combination of the persistence length and a diffusive length scale for thermal systems. This new insight, for which we also provide a simple physical argument, allows us to reconcile and explain the manifestly disparate departures from equilibrium reported in many previous studies of dense active materials.

Correlated escape of active particles across a potential barrier

We study the dynamics of one-dimensional active particles confined in a double-well potential, focusing on the escape properties of the system, such as the mean escape time from a well. We first consider a single-particle both in near and far-from-equilibrium regimes by varying the persistence time of the active force and the swim velocity. A non-monotonic behavior of the mean escape time is observed with the persistence time of the activity, revealing the existence of an optimal choice of the parameters favoring the escape process. For small persistence times, a Kramers-like formula with an effective potential obtained within the unified colored noise approximation is shown to hold. Instead, for large persistence times, we developed a simple theoretical argument based on the first passage theory, which explains the linear dependence of the escape time with the persistence of the active force. In the second part of the work, we consider the escape on two active particles mutually repelling. Interestingly, the subtle interplay of active and repulsive forces may lead to a correlation between particles, favoring the simultaneous jump across the barrier. This mechanism cannot be observed in the escape process of two passive particles. Finally, we find that in the small persistence regime, the repulsion favors the escape, such as in passive systems, in agreement with our theoretical predictions, while for large persistence times, the repulsive and active forces produce an effective attraction, which hinders the barrier crossing.

Alignment interactions drive structural transitions in biological tissues

Experimental evidence shows that there is a feedback between cell shape and cell motion. How this feedback impacts the collective behavior of dense cell monolayers remains an open question. We investigate the effect of a feedback that tends to align the cell crawling direction with cell elongation in a biological tissue model. We find that the alignment interaction promotes nematic patterns in the fluid phase that eventually undergo a nonequilibrium phase transition into a quasihexagonal solid. Meanwhile, highly asymmetric cells do not undergo the liquid-to-solid transition for any value of the alignment coupling. In this regime, the dynamics of cell centers and shape fluctuation show features typical of glassy systems.

Shear-induced orientational ordering in an active glass former

Dense assemblies of self-propelled particles that can form solid-like states also known as active or living glasses are abundant around us, covering a broad range of length scales and timescales: from the cytoplasm to tissues, from bacterial biofilms to vehicular traffic jams, and from Janus colloids to animal herds. Being structurally disordered as well as strongly out of equilibrium, these systems show fascinating dynamical and mechanical properties. Using extensive molecular dynamics simulation and a number of distinct dynamical and mechanical order parameters, we differentiate three dynamical steady states in a sheared model active glassy system: 1) a disordered state, 2) a propulsion-induced ordered state, and 3) a shear-induced ordered state. We supplement these observations with an analytical theory based on an effective single-particle Fokker-Planck description to rationalize the existence of the shear-induced orientational ordering behavior in an active glassy system without explicit aligning interactions of, for example, Vicsek type. This ordering phenomenon occurs in the large persistence time limit and is made possible only by the applied steady shear. Using a Fokker-Planck description with parameters that can be measured independently, we make testable predictions for the joint distribution of single-particle position and orientation. These predictions match well with the joint distribution measured from direct numerical simulation. Our results are of relevance for experiments exploring the rheological response of dense active colloids and jammed active granular matter systems.

Capillary Interfacial Tension in Active Phase Separation

In passive fluid-fluid phase separation, a single interfacial tension sets both the capillary fluctuations of the interface and the rate of Ostwald ripening. We show that these phenomena are governed by two different tensions in active systems, and compute the capillary tension σ_{cw} which sets the relaxation rate of interfacial fluctuations in accordance with capillary wave theory. We discover that strong enough activity can cause negative σ_{cw}. In this regime, depending on the global composition, the system self-organizes, either into a microphase-separated state in which coalescence is highly inhibited, or into an "active foam" state. Our results are obtained for Active Model B+, a minimal continuum model which, although generic, admits significant analytical progress.

Optimization of an active heat engine

Optimization of heat engines at the microscale has applications in biological and artificial nanotechnology and stimulates theoretical research in nonequilibrium statistical physics. Here we consider noninteracting overdamped particles confined by an external harmonic potential, in contact with either a thermal reservoir or a stochastic self-propulsion force (active Ornstein-Uhlenbeck model). A cyclical machine is produced by periodic variation of the parameters of the potential and of the noise. An exact mapping between the passive and the active model allows us to define the effective temperature T_{eff}(t), which is meaningful for the thermodynamic performance of the engine. We show that T_{eff}(t) is different from all other known active temperatures, typically used in static situations. The mapping allows us to optimize the active engine, regardless of the values of the persistence time or self-propulsion velocity. In particular, through linear irreversible thermodynamics (small amplitude of the cycle), we give an explicit formula for the optimal cycle period and phase delay (between the two modulated parameters, stiffness and temperature) achieving maximum power with Curzon-Ahlborn efficiency. In the quasistatic limit, the formula for T_{eff}(t) simplifies and coincides with a recently proposed temperature for stochastic thermodynamics, bearing a compact expression for the maximum efficiency. A point, which has been overlooked in recent literature, is made about the difficulty in defining efficiency without a consistent definition of effective temperature.

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.

Dynamical properties of different models of elastic polymer rings: Confirming the link between deformation and fragility

We report extensive numerical simulations of different models of 2D polymer rings with internal elasticity. We monitor the dynamical behavior of the rings as a function of the packing fraction to address the effects of particle deformation on the collective response of the system. In particular, we compare three different models: (i) a recently investigated model [N. Gnan and E. Zaccarelli, Nat. Phys. 15, 683 (2019)] where an inner Hertzian field providing the internal elasticity acts on the monomers of the ring, (ii) the same model where the effect of such a field on the center of mass is balanced by opposite forces, and (iii) a semi-flexible model where an angular potential between adjacent monomers induces strong particle deformations. By analyzing the dynamics of the three models, we find that in all cases, there exists a direct link between the system fragility and particle asphericity. Among the three, only the first model displays anomalous dynamics in the form of a super-diffusive behavior of the mean-squared displacement and of a compressed exponential relaxation of the density auto-correlation function. We show that this is due to the combination of internal elasticity and the out-of-equilibrium force self-generated by each ring, both of which are necessary ingredients to induce such a peculiar behavior often observed in experiments of colloidal gels. These findings reinforce the role of particle deformation, connected to internal elasticity, in driving the dynamical response of dense soft particles.

Universality class of the motility-induced critical point in large scale off-lattice simulations of active particles

We perform large-scale computer simulations of an off-lattice two-dimensional model of active particles undergoing a motility-induced phase separation (MIPS) to investigate the system's critical behaviour close to the critical point of the MIPS curve. By sampling steady-state configurations for large system sizes and performing finite size scaling analysis we provide exhaustive evidence that the critical behaviour of this active system belongs to the Ising universality class. In addition to the scaling observables that are also typical of passive systems, we study the critical behaviour of the kinetic temperature difference between the two active phases. This quantity, which is always zero in equilibrium, displays instead a critical behavior in the active system which is well described by the same exponent of the order parameter in agreement with mean-field theory.

Statistical mechanics of active Ornstein-Uhlenbeck particles

We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.

Fluctuation–Dissipation Relations in Active Matter Systems

We investigate the non-equilibrium character of self-propelled particles through the study of the linear response of the active Ornstein–Uhlenbeck particle (AOUP) model. We express the linear response in terms of correlations computed in the absence of perturbations, proposing a particularly compact and readable fluctuation–dissipation relation (FDR): such an expression explicitly separates equilibrium and non-equilibrium contributions due to self-propulsion. As a case study, we consider non-interacting AOUP confined in single-well and double-well potentials. In the former case, we also unveil the effect of dimensionality, studying one-, two-, and three-dimensional dynamics. We show that information about the distance from equilibrium can be deduced from the FDR, putting in evidence the roles of position and velocity variables in the non-equilibrium relaxation.

Fluctuating Motion in an Active Environment

We derive the fluctuation dynamics of a probe in weak coupling with a living medium, modeled as particles undergoing an active Ornstein-Uhlenbeck dynamics. Nondissipative corrections to the fluctuation-dissipation relation are written out explicitly in terms of time correlations in the active medium. A first term changes the inertial mass of the probe as a consequence of the persistence of the active medium. A second correction modifies the friction kernel. The resulting generalized Langevin equation benchmarks the motion induced on probes immersed in active versus passive media. The derivation uses nonequilibrium response theory.

Negative friction memory induces persistent motion

We investigate the mean-square displacement (MSD) for random motion governed by the generalized Langevin equation for memory functions that contain two different time scales: In the first model, the memory kernel consists of a delta peak and a single-exponential and in the second model of the sum of two exponentials. In particular, we investigate the scenario where the long-time exponential kernel contribution is negative. The competition between positive and negative friction memory contributions produces an enhanced transient persistent regime in the MSD, which is relevant for biological motility and active matter systems.

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.

Single active particle engine utilizing a nonreciprocal coupling between particle position and self-propulsion

We recently argued that a self-propelled particle is formally equivalent to a system consisting of two subsystems coupled by a nonreciprocal interaction [Phys. Rev. E 100, 050603(R) (2019)2470-004510.1103/PhysRevE.100.050603]. Here, we show that this nonreciprocal coupling allows us to extract useful work from a single self-propelled particle maintained at constant temperature, by using an aligning interaction to control correlations between the particle's position and self-propulsion.

Active matter: Quantifying the departure from equilibrium

Active matter systems are driven out of equilibrium at the level of individual constituents. One widely studied class are systems of athermal particles that move under the combined influence of interparticle interactions and self-propulsions, with the latter evolving according to the Ornstein-Uhlenbeck stochastic process. Intuitively, these so-called active Ornstein-Uhlenbeck particle (AOUP) systems are farther from equilibrium for longer self-propulsion persistence times. Quantitatively, this is confirmed by the increasing equal-time velocity correlations (which are trivial in equilibrium) and by the increasing violation of the Einstein relation between the self-diffusion and mobility coefficients. In contrast, the entropy production rate, calculated from the ratio of the probabilities of the position space trajectory and its time-reversed counterpart, has a nonmonotonic dependence on the persistence time. Thus, it does not properly quantify the departure of AOUP systems from equilibrium.

Entropy production of an active particle in a box

A run-and-tumble particle in a one-dimensional box (infinite potential well) is studied. The steady state is analytically solved and analyzed, revealing the emergent length scale of the boundary layer where particles accumulate near the walls. The mesoscopic steady state entropy production rate of the system is derived from coupled Fokker-Planck equations with a linear reaction term, resulting in an exact analytic expression. The entropy production density is shown to peak at the walls. Additionally, the derivative of the entropy production rate peaks at a system size proportional to the length scale of the accumulation boundary layer, suggesting that the behavior of the entropy production rate and its derivatives as a function of the control parameter may signify a qualitative behavior change in the physics of active systems, such as phase transitions.

Thermodynamic cycles with active matter

Active matter constantly dissipates energy to power the self-propulsion of its microscopic constituents. This opens the door to designing innovative cyclic engines without any equilibrium equivalent. We offer a consistent thermodynamic framework to characterize and optimize the performances of such cycles. Based on a minimal model, we put forward a protocol which extracts work by controlling only the properties of the confining walls at boundaries, and we rationalize the transitions between optimal cycles. We show that the corresponding power and efficiency are generally proportional, so that they reach their maximum values at the same cycle time in contrast with thermal cycles, and we provide a generic relation constraining the fluctuations of the power.

Diffusion properties of self-propelled particles in cellular flows

We study the dynamics of a self-propelled particle advected by a steady laminar flow. The persistent motion of the self-propelled particle is described by an active Ornstein-Uhlenbeck process. We focus on the diffusivity properties of the particle as a function of persistence time and free-diffusion coefficient, revealing non-monotonic behaviors, with the occurrence of a minimum and a steep growth in the regime of large persistence time. In the latter limit, we obtain an analytical prediction for the scaling of the diffusion coefficient with the parameters of the active force. Our study sheds light on the effect of a flow-field on the diffusion of active particles, such as living microorganisms and motile phytoplankton in fluids.

Three-body problem for Langevin dynamics with different temperatures

A mixture of Brownian particles at different temperatures has been a useful model for studying the out-of-equilibrium properties of systems made up of microscopic components with differing levels of activity. This model was previously studied analytically for two-particle interactions in the dilute limit, yielding a Boltzmann-like two-particle distribution with an effective temperature. Like the Newtonian two- and three-body problems, we ask here whether the two-particle results can be extended to three-particle interactions to get the three-particle distributions. By considering the special solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution cannot in general be Boltzmann-like with an effective temperature. We instead find that the steady-state distribution of any two particles in a triplet depends on the properties of and interactions with the third particle, leading to some unexpected behaviors not present in equilibrium.

Collective forces in scalar active matter

Large-scale collective behavior in suspensions of active particles can be understood from the balance of statistical forces emerging beyond the direct microscopic particle interactions. Here we review some aspects of the collective forces that can arise in suspensions of self-propelled active Brownian particles: wall forces under confinement, interfacial forces, and forces on immersed bodies mediated by the suspension. Even for non-aligning active particles, these forces are intimately related to a non-uniform polarization of particle orientations induced by walls and bodies, or inhomogeneous density profiles. We conclude by pointing out future directions and promising areas for the application of collective forces in synthetic active matter, as well as their role in living active matter.

Two-dimensional active motion

The diffusion in two dimensions of noninteracting active particles that follow an arbitrary motility pattern is considered for analysis. A Fokker-Planck-like equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of active motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided, and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. The analysis is focused in the intermediate-time regime, where the effects of the specific pattern of active motion are conspicuous. For this, it is shown that only the expectation value of the first two harmonics of the scattering angle of the direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.

Nitrate reduction in the reed rhizosphere of a riparian zone: From functional genes to activity and contribution

The increased nitrogen (N) fertilizer usage caused substantial nitrate (NO₃^-) leaching into groundwater and eutrophication in downstream aquatic systems. Riparian zones positioned as the link interfaces of terrestrial and aquatic environments are effective in NO₃^- removal. However, the microbial mechanisms regulating NO₃^- reduction in riparian zones are still unclear. In this study, four microbial NO₃^- reduction processes were explored in fine-scale riparian soil horizons by isotopic tracing technique, qPCR of functional gene, high-throughput amplicon sequencing, and phylogenetic molecular ecological network analysis. Interestingly, anaerobic ammonium oxidation (anammox) contributed to NO₃^- removal of up to 48.2% only in waterward sediments but not in landward soil. Denitrification was still the most significant contributor to NO₃^- reduction (32.0-91.8%) and N-losses (51.7-100%). Additionally, dissimilatory nitrate reduction to ammonium (DNRA) played a key role in NO₃^- reduction (4.4-67.5%) and was even comparable to denitrification. Community structure analysis of denitrifying, anammox, and DNRA bacterial communities targeting the related functional gene showed that spatial heterogeneity played a greater role than both temporal and soil type (rhizosphere and non-rhizosphere soil) variability in microbial community structuring. Denitrification and DNRA communities were diverse, and their activities did not depend on gene abundance but were significantly related to organic matter, suggesting that gene abundance alone was insufficient in assessing their activity in riparian zones. Based on networks, DNRA plays a keystone role among the microbial NO₃^- reducers. As the last line of defense in the interception of terrestrial NO₃^-, these findings contribute to our understanding of NO₃^- removal mechanisms in riparian zones, and could potentially be exploited to reduce the diffusion of NO₃^- pollution.