Effective temperature of active matter

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Statistics of Colloidal Suspensions Stirred by Microswimmers

We present a statistical analysis of the experimental trajectories of colloids in a dilute suspension of the green algae Chlamydomonas reinhardtii. The measured probability density function (pdf) of the displacements of colloids covers 7 orders of magnitude. The pdfs are characterized by non-Gaussian tails for intermediate time intervals, but nevertheless they collapse when scaled with their standard deviation. This diffusive scaling breaks down for longer time intervals and the pdf becomes Gaussian. However, the mean squared displacements of tracer positions are linear over the complete measurement time interval. Experiments are performed for various tracer diameters, swimmer concentrations, and mean swimmer velocities. This allows a rigorous comparison with several theoretical models. We can exclude a description based on an effective temperature and other mean field approaches that describe the irregular motion as a sum of the fluctuating far field of many microswimmers. The data are best described by the microscopic model by J.-L. Thiffeault, Distribution of particle displacements due to swimming microorganisms, Phys. Rev. E 92, 023023 (2015)PRESCM1539-375510.1103/PhysRevE.92.023023.

Erratum to: Effective temperature of active fluids and sheared soft glassy materials

The authors were notified by their collaborators Golan Bel and Dan Wexler that the expression for the over-damped limit of the kinetic energy of the trapped particle, that they have used in the orignal paper, was in error. They provide the correct expressions in this erratum.

Signatures of motor susceptibility to forces in the dynamics of a tracer particle in an active gel

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

Stationary superstatistics distributions of trapped run-and-tumble particles

We present an analysis of the stationary distributions of run-and-tumble particles trapped in external potentials in terms of a thermophoretic potential that emerges when trapped active motion is mapped to trapped passive Brownian motion in a fictitious inhomogeneous thermal bath. We elaborate on the meaning of the non-Boltzmann-Gibbs stationary distributions that emerge as a consequence of the persistent motion of active particles. These stationary distributions are interpreted as a class of distributions in nonequilibrium statistical mechanics known as superstatistics. Our analysis provides an original insight on the link between the intrinsic nonequilibrium nature of active motion and the well-known concept of local equilibrium used in nonequilibrium statistical mechanics and contributes to the understanding of the validity of the concept of effective temperature. Particular cases of interest, regarding specific trapping potentials used in other theoretical or experimental studies, are discussed. We point out as an unprecedented effect, the emergence of new modes of the stationary distribution as a consequence of the coupling of persistent motion in a trapping potential that varies highly enough with position.

Biological Tissues as Active Nematic Liquid Crystals

Live tissues can self-organize and be described as active materials composed of cells that generate active stresses through continuous injection of energy. In vitro reconstituted molecular networks, as well as single-cell cytoskeletons show that their filamentous structures can portray nematic liquid crystalline properties and can promote nonequilibrium processes induced by active processes at the microscale. The appearance of collective patterns, the formation of topological singularities, and spontaneous phase transition within the cell cytoskeleton are emergent properties that drive cellular functions. More integrated systems such as tissues have cells that can be seen as coarse-grained active nematic particles and their interaction can dictate many important tissue processes such as epithelial cell extrusion and migration as observed in vitro and in vivo. Here, a brief introduction to the concept of active nematics is provided, and the main focus is on the use of this framework in the systematic study of predominantly 2D tissue architectures and dynamics in vitro. In addition how the nematic state is important in tissue behavior, such as epithelial expansion, tissue homeostasis, and the atherosclerosis disease state, is discussed. Finally, how the nematic organization of cells can be controlled in vitro for tissue engineering purposes is briefly discussed.

Active dumbbells: Dynamics and morphology in the coexisting region

With the help of molecular dynamics simulations we study an ensemble of active dumbbells in purely repulsive interaction. We derive the phase diagram in the density-activity plane and we characterise the various phases with liquid, hexatic and solid character. The analysis of the structural and dynamical properties, such as enstrophy, mean-square displacement, polarisation, and correlation functions, shows the continuous character of liquid and hexatic phases in the coexisting region when the activity is increased starting from the passive limit.

Effective temperature of active fluids and sheared soft glassy materials

The dynamics within active fluids, driven by internal activity of the self-propelled particles, is a subject of intense study in non-equilibrium physics. These systems have been explored using simulations, where the motion of a passive tracer particle is followed. Similar studies have been carried out for a soft glassy material that is driven by shearing its boundaries. In both types of systems the non-equilibrium motion have been quantified by defining a set of "effective temperatures", using both the tracer particle kinetic energy and the fluctuation-dissipation relation. We demonstrate that these effective temperatures extracted from the many-body simulations fit analytical expressions that are obtained for a single active particle inside a visco-elastic fluid. This result provides testable predictions and suggests a unified description for the dynamics inside active systems.

Living Matter: Mesoscopic Active Materials

An introduction to the physical properties of living active matter at the mesoscopic scale (tens of nanometers to micrometers) and their unique features compared with "dead," nonactive matter is presented. This field of research is increasingly denoted as "biological physics" where physics includes chemical physics, soft matter physics, hydrodynamics, mechanics, and the related engineering sciences. The focus is on the emergent properties of these systems and their collective behavior, which results in active self-organization and how they relate to cellular-level biological function. These include locomotion (cell motility and migration) forces that give rise to cell division, the growth and form of cellular assemblies in development, the beating of heart cells, and the effects of mechanical perturbations such as shear flow (in the bloodstream) or adhesion to other cells or tissues. An introduction to the fundamental concepts and theory with selected experimental examples related to the authors' own research is presented, including red-blood-cell membrane fluctuations, motion of the nucleus within an egg cell, self-contracting acto-myosin gels, and structure and beating of heart cells (cardiomyocytes), including how they can be driven by an oscillating, mechanical probe.

Hidden entropy production and work fluctuations in an ideal active gas

Collections of self-propelled particles that move persistently by continuously consuming free energy are a paradigmatic example of active matter. In these systems, unlike Brownian "hot colloids," the breakdown of detailed balance yields a continuous production of entropy at steady state, even for an ideal active gas. We quantify the irreversibility for a noninteracting active particle in two dimensions by treating both conjugated and time-reversed dynamics. By starting with underdamped dynamics, we identify a hidden rate of entropy production required to maintain persistence and prevent the rapidly relaxing momenta from thermalizing, even in the limit of very large friction. Additionally, comparing two popular models of self-propulsion with identical dissipation on average, we find that the fluctuations and large deviations in work done are markedly different, providing thermodynamic insight into the varying extents to which macroscopically similar active matter systems may depart from equilibrium.

Cargo carrying bacteria at interfaces

The displacements of ensembles of colloids at the interface between oil and suspensions of the bacterium Pseudomonas aeruginosa PA14ΔpelA indicate enhanced colloid mobilities and apparently diffusive motion driven by interactions with the bacteria. However, inspection of individual trajectories of ∼500 particles reveals prolonged, directed displacements inconsistent with purely hydrodynamic interactions between swimming bacteria and colloids. Analysis of the properties of colloid paths indicates trajectories can be sorted into four distinct categories, including diffusive, persistent, curly, and mixed trajectory types. Non-diffusive trajectories are the norm, comprising 2/3 of the observed trajectories. Imaging of colloids in the interface reveals anisotropic assemblies formed by colloids decorated with one or more adhered bacteria that drive the colloids along these paths. The trajectories and enhanced transport result from individual colloids being moved as cargo by these adhered bacteria. The implications of these structures and open questions for interfacial transport are discussed and related to the active colloid literature.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

Effective interactions and dynamics of small passive particles in an active bacterial medium

This article presents an investigation of the interparticle interactions and dynamics of submicron silica colloids suspended in a bath of motile Escherichia coli bacteria. The colloidal microstructure and dynamics were probed by ultra-small-angle x-ray scattering and multi-speckles x-ray photon correlation spectroscopy, respectively. Both static and hydrodynamic interactions were obtained for different colloid volume fractions and bacteria concentrations as well as when the interparticle interaction potential was modified by the motility buffer. Results suggest that motile bacteria reduce the effective attractive interactions between passive colloids and enhance their dynamics at high colloid volume fractions. The enhanced dynamics under different static interparticle interactions can be rationalized in terms of an effective viscosity of the medium and unified by means of an empirical effective temperature of the system. While the influence of swimming bacteria on the colloid dynamics is significantly lower for small particles, the role of motility buffer on the static and dynamic interactions becomes more pronounced.

Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an active bath

We investigate experimentally and numerically the stochastic dynamics and the time-dependent response of colloids subject to a small external perturbation in a dense bath of motile E. coli bacteria. The external field is a magnetic field acting on a superparamagnetic microbead suspended in an active medium. The measured linear response reveals an instantaneous friction kernel despite the complexity of the bacterial bath. By comparing the mean squared displacement and the response function we detect a clear violation of the fluctuation dissipation theorem.

Active Brownian equation of state: metastability and phase coexistence

As a result of the competition between self-propulsion and excluded volume interactions, purely repulsive self-propelled spherical particles undergo a motility-induced phase separation (MIPS). We carry out a systematic computational study, considering several interaction potentials, systems confined by hard walls or with periodic boundary conditions, and different initial conditions. This approach allows us to identify that, despite its non-equilibrium nature, the equations of state of Active Brownian Particles (ABP) across MIPS verify the characteristic properties of first-order liquid-gas phase transitions, meaning, equality of pressure of the coexisting phases once a nucleation barrier has been overcome and, in the opposite case, hysteresis around the transition as long as the system remains in the metastable region. Our results show that the equations of state of ABPs account for their phase behaviour, providing a firm basis to describe MIPS as an equilibrium-like phase transition.

Large Fluctuations and Dynamic Phase Transition in a System of Self-Propelled Particles

We study the statistics, in stationary conditions, of the work W_{τ} done by the active force in different systems of self-propelled particles in a time τ. We show the existence of a critical value W_{τ}^{†} such that fluctuations with W_{τ}>W_{τ}^{†} correspond to configurations where interaction between particles plays a minor role whereas those with W_{τ}<W_{τ}^{†} represent states with single particles dragged by clusters. This twofold behavior is fully mirrored by the probability distribution P(W_{τ}) of the work, which does not obey the large-deviation principle for W_{τ}<W_{τ}^{†}. This pattern of behavior can be interpreted as due to a phase transition occurring at the level of fluctuating quantities and an order parameter is correspondingly identified.

Films of bacteria at interfaces

Bacteria are often discussed as active colloids, self-propelled organisms whose collective motion can be studied in the context of non-equilibrium statistical mechanics. In such studies, the behavior of bacteria confined to interfaces or in the proximity of an interface plays an important role. For instance, many studies have probed collective behavior of bacteria in quasi two-dimensional systems such as soap films. Since fluid interfaces can adsorb surfactants and other materials, the stress and velocity boundary conditions at interfaces can alter bacteria motion; hydrodynamic studies of interfaces with differing boundary conditions are reviewed. Also, bacteria in bulk can become trapped at or near fluid interfaces, where they colonize and form structures comprising secretions like exopolysaccharides, surfactants, living and dead bacteria, thereby creating Films of Bacteria at Interfaces (FBI). The formation of FBI is discussed at air-water, oil-water, and water-water interfaces, with an emphasis on film mechanics, and with some allusion to genetic functions guiding bacteria to restructure fluid interfaces. At air-water interfaces, bacteria form pellicles or interfacial biofilms. Studies are reviewed that reveal that pellicle material properties differ for different strains of bacteria, and that pellicle physicochemistry can act as a feedback mechanism to regulate film formation. At oil-water interfaces, a range of FBI form, depending on bacteria strain. Some bacteria-laden interfaces age from an initial active film, with dynamics dominated by motile bacteria, through viscoelastic states, to form an elastic film. Others remain active with no evidence of elastic film formation even at significant interface ages. Finally, bacteria can adhere to and colonize ultra-low surface tension interfaces such as aqueous-aqueous systems common in food industries. Relevant literature is reviewed, and areas of interest for potential application are discussed, ranging from health to bioremediation.

Nonequilibrium steady states in Langevin thermal systems

Equilibrium is characterized by its fundamental properties, such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relation mutually exclude each other, and no equivalence relation is possible between any two of the three properties. This implies that a steady state of Langevin systems with velocity-dependent forces may maintain some equilibrium properties but not all of them. Our results are illustrated with a few example systems.

Driven Brownian particle as a paradigm for a nonequilibrium heat bath: Effective temperature and cyclic work extraction

We apply the concept of a frequency-dependent effective temperature based on the fluctuation-dissipation ratio to a driven Brownian particle in a nonequilibrium steady state. Using this system as a thermostat for a weakly coupled harmonic oscillator, the oscillator thermalizes according to a canonical distribution at the respective effective temperature across the entire frequency spectrum. By turning the oscillator from a passive thermometer into a heat engine, we realize the cyclic extraction of work from a single thermal reservoir, which is feasible only due to its nonequilibrium nature.

Crystallization and flow in active patch systems

Based upon recent experiments in which Janus particles are made into active swimmers by illuminating them with laser light, we explore the effect of applying a light pattern on the sample, thereby creating activity inducing zones or active patches. We simulate a system of interacting Brownian diffusers that become active swimmers when moving inside an active patch and analyze the structure and dynamics of the ensuing stationary state. We find that, in some respects, the effect of spatially inhomogeneous activity is qualitatively similar to a temperature gradient. For asymmetric patches, however, this analogy breaks down because the ensuing stationary state is specific to partial active motion.

Transient Casimir Forces from Quenches in Thermal and Active Matter

We compute fluctuation-induced (Casimir) forces for classical systems after a temperature quench. Using a generic coarse-grained model for fluctuations of a conserved density, we find that transient forces arise even if the initial and final states are force free. In setups reminiscent of Casimir (planar walls) and van der Waals (small inclusions) interactions, we find comparable exact universal expressions for the force. Dynamical details only scale the time axis of transient force curves. We propose that such quenches can be achieved, for instance, in experiments on active matter, employing tunable activity or interaction protocols.

Glassy dynamics of Brownian particles with velocity-dependent friction

We consider a two-dimensional model system of Brownian particles in which slow particles are accelerated while fast particles are damped. The motion of the individual particles is described by a Langevin equation with Rayleigh-Helmholtz velocity-dependent friction. In the case of noninteracting particles, the time evolution equations lead to a non-Gaussian velocity distribution. The velocity-dependent friction allows negative values of the friction or energy intakes by slow particles, which we consider active motion, and also causes breaking of the fluctuation dissipation relation. Defining the effective temperature proportional to the second moment of velocity, it is shown that for a constant effective temperature the higher the noise strength, the lower the number of active particles in the system. Using the Mori-Zwanzig formalism and the mode-coupling approximation, the equations of motion for the density autocorrelation function are derived. The equations are solved using the equilibrium structure factors. The integration-through-transients approach is used to derive a relation between the structure factor in the stationary state considering the interacting forces, and the conventional equilibrium static structure factor.

Linear response to leadership, effective temperature, and decision making in flocks

Large collections of autonomously moving agents, such as animals or micro-organisms, are able to flock coherently in space even in the absence of a central control mechanism. While the direction of the flock resulting from this critical behavior is random, this can be controlled by a small subset of informed individuals acting as leaders of the group. In this article we use the Vicsek model to investigate how flocks respond to leadership and make decisions. Using a combination of numerical simulations and continuous modeling we demonstrate that flocks display a linear response to leadership that can be cast in the framework of the fluctuation-dissipation theorem, identifying an effective temperature reflecting how promptly the flock reacts to the initiative of the leaders. The linear response to leadership also holds in the presence of two groups of informed individuals with competing interests, indicating that the flock's behavioral decision is determined by both the number of leaders and their degree of influence.

How Far from Equilibrium Is Active Matter?

Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relation between injection and dissipation of energy at the microscopic scale. We consider such a system of interacting particles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics still satisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that, for short persistent times, the entropy production rate vanishes. This endows such systems with an effective fluctuation-dissipation theorem akin to that of thermal equilibrium systems. Last, we show how interacting particle systems with viscous drags and correlated noises can be seen as in equilibrium with a viscoelastic bath but driven out of equilibrium by nonconservative forces, hence providing energetic insight into the departure of active systems from equilibrium.

Configurational entropy and effective temperature in systems of active Brownian particles

We propose a method to determine the effective density of states and configurational entropy in systems of active Brownian particles by measuring the probability distribution function of potential energy at varying temperatures. Assuming that the entropy is a continuous and monotonically increasing function of energy, we provide support that two-dimensional systems of purely repulsive active Brownian spheres can be mapped onto an equilibrium system with a Boltzmann-like distribution and an effective temperature. We find that the effective temperature depends even for a large number of particles on system size, suggesting that active systems are non-extensive. In addition, the effective Helmholtz free energy can be derived from the configurational entropy. We verify our results regarding the configurational entropy by using thermodynamic integration of the effective Helmholtz free energy with respect to temperature.

Particle diffusion in active fluids is non-monotonic in size

We experimentally investigate the effect of particle size on the motion of passive polystyrene spheres in suspensions of Escherichia coli. Using particles covering a range of sizes from 0.6 to 39 microns, we probe particle dynamics at both short and long time scales. In all cases, the particles exhibit super-diffusive ballistic behavior at short times before eventually transitioning to diffusive behavior. Surprisingly, we find a regime in which larger particles can diffuse faster than smaller particles: the particle long-time effective diffusivity exhibits a peak in particle size, which is a deviation from classical thermal diffusion. We also find that the active contribution to particle diffusion is controlled by a dimensionless parameter, the Péclet number. A minimal model qualitatively explains the existence of the effective diffusivity peak and its dependence on bacterial concentration. Our results have broad implications on characterizing active fluids using concepts drawn from classical thermodynamics.

Rotational and translational diffusion in an interacting active dumbbell system

We study the dynamical properties of a two-dimensional ensemble of self-propelled dumbbells with only repulsive interactions. This model undergoes a phase transition between a homogeneous and a segregated phase and we focus on the former. We analyze the translational and rotational mean-square displacements in terms of the Péclet number, describing the relative role of active forces and thermal fluctuations, and of particle density. We find that the four distinct regimes of the translational mean-square displacement of the single active dumbbell survive at finite density for parameters that lead to a separation of time scales. We establish the Péclet number and density dependence of the diffusion constant in the last diffusive regime. We prove that the ratio between the diffusion constant and its value for the single dumbbell depends on temperature and active force only through the Péclet number at all densities explored. We also study the rotational mean-square displacement proving the existence of a rich behavior with intermediate regimes only appearing at finite density. The ratio of the rotational late-time diffusion constant and its vanishing density limit depends on the Péclet number and density only. At low Péclet number it is a monotonically decreasing function of density. At high Péclet number it first increases to reach a maximum and then decreases as a function of density. We interpret the latter result advocating the presence of large-scale fluctuations close to the transition, at large-enough density, that favor coherent rotation inhibiting, however, rotational motion for even larger packing fractions.

Substrate stiffness-modulated registry phase correlations in cardiomyocytes map structural order to coherent beating

Recent experiments show that both striation, an indication of the structural registry in muscle fibres, as well as the contractile strains produced by beating cardiac muscle cells can be optimized by substrate stiffness. Here we show theoretically how the substrate rigidity dependence of the registry data can be mapped onto that of the strain measurements. We express the elasticity-mediated structural registry as a phase-order parameter using a statistical physics approach that takes the noise and disorder inherent in biological systems into account. By assuming that structurally registered myofibrils also tend to beat in phase, we explain the observed dependence of both striation and strain measurements of cardiomyocytes on substrate stiffness in a unified manner. The agreement of our ideas with experiment suggests that the correlated beating of heart cells may be limited by the structural order of the myofibrils, which in turn is regulated by their elastic environment.

Dynamics of a homogeneous active dumbbell system

We analyze the dynamics of a two-dimensional system of interacting active dumbbells. We characterize the mean-square displacement, linear response function, and deviation from the equilibrium fluctuation-dissipation theorem as a function of activity strength, packing fraction, and temperature for parameters such that the system is in its homogeneous phase. While the diffusion constant in the last diffusive regime naturally increases with activity and decreases with packing fraction, we exhibit an intriguing nonmonotonic dependence on the activity of the ratio between the finite-density and the single-particle diffusion constants. At fixed packing fraction, the time-integrated linear response function depends nonmonotonically on activity strength. The effective temperature extracted from the ratio between the integrated linear response and the mean-square displacement in the last diffusive regime is always higher than the ambient temperature, increases with increasing activity, and, for small active force, monotonically increases with density while for sufficiently high activity it first increases and next decreases with the packing fraction. We ascribe this peculiar effect to the existence of finite-size clusters for sufficiently high activity and density at the fixed (low) temperatures at which we worked. The crossover occurs at lower activity or density the lower the external temperature. The finite-density effective temperature is higher (lower) than the single dumbbell one below (above) a crossover value of the Péclet number.

Reentrance in an active glass mixture

Active matter, whose motion is driven, and glasses, whose dynamics are arrested, seem to lie at opposite ends of the spectrum in nonequilibrium systems. In spite of this, both classes of systems exhibit a multitude of stable states that are dynamically isolated from one another. While this defining characteristic is held in common, its origin is different in each case: for active systems, the irreversible driving forces can produce dynamically frozen states, while glassy systems vitrify when they get kinetically trapped on a rugged free energy landscape. In a mixture of active and glassy particles, the interplay between these two tendencies leads to novel phenomenology. We demonstrate this with a spin glass model that we generalize to include an active component. In the absence of a ferromagnetic bias, we find that the spin glass transition temperature depresses with the active fraction, consistent with what has been observed for fully active glassy systems. When a bias does exist, however, a new type of transition becomes possible: the system can be cooled out of the glassy phase. This unusual phenomenon, known as reentrance, has been observed before in a limited number of colloidal and micellar systems, but it has not yet been observed in active glass mixtures. Using low order perturbation theory, we study the origin of this reentrance and, based on the physical picture that results, suggest how our predictions might be measured experimentally.

Realization of nonequilibrium thermodynamic processes using external colored noise

We investigate the dynamics of single microparticles immersed in water that are driven out of equilibrium in the presence of an additional external colored noise. As a case study, we trap a single polystyrene particle in water with optical tweezers and apply an external electric field with flat spectrum but a finite bandwidth of the order of kHz. The intensity of the external noise controls the amplitude of the fluctuations of the position of the particle and therefore of its effective temperature. Here we show, in two different nonequilibrium experiments, that the fluctuations of the work done on the particle obey the Crooks fluctuation theorem at the equilibrium effective temperature, given that the sampling frequency and the noise cutoff frequency are properly chosen.

Clustering and heterogeneous dynamics in a kinetic Monte Carlo model of self-propelled hard disks

We introduce a kinetic Monte Carlo model for self-propelled hard disks to capture with minimal ingredients the interplay between thermal fluctuations, excluded volume, and self-propulsion in large assemblies of active particles. We analyze in detail the resulting (density, self-propulsion) nonequilibrium phase diagram over a broad range of parameters. We find that purely repulsive hard disks spontaneously aggregate into fractal clusters as self-propulsion is increased and rationalize the evolution of the average cluster size by developing a kinetic model of reversible aggregation. As density is increased, the nonequilibrium clusters percolate to form a ramified structure reminiscent of a physical gel. We show that the addition of a finite amount of noise is needed to trigger a nonequilibrium phase separation, showing that demixing in active Brownian particles results from a delicate balance between noise, interparticle interactions, and self-propulsion. We show that self-propulsion has a profound influence on the dynamics of the active fluid. We find that the diffusion constant has a nonmonotonic behavior as self-propulsion is increased at finite density and that activity produces strong deviations from Fickian diffusion that persist over large time scales and length scales, suggesting that systems of active particles generically behave as dynamically heterogeneous systems.

Diffusion, sedimentation equilibrium, and harmonic trapping of run-and-tumble nanoswimmers

The diffusion of self-propelling nanoswimmers is explored by dissipative particle dynamics in which a nanoswimmer swims by forming an instantaneous force dipole with one of its nearest neighboring solvent beads. Our simulations mimic run-and-tumble behavior by letting the swimmer run for a time τ, then it randomly changes its direction for the next run period. Our simulations show that the swimming speed (ν(a)) of a nanoswimmer is proportional to the propulsion force and the mobility of a pusher is the same as that of a puller. The effective diffusivity is determined by three methods: mean squared displacement, velocity autocorrelation function, and sedimentation equilibrium. The active colloid undergoes directed propulsion at short time scales but changes to random motion at long time scales. The velocity autocorrelation function decreases with time and becomes zero beyond the run time. Under gravity, the concentration profile of active colloids follows Boltzmann distribution with a sedimentation length consistent with that acquired from the drift-diffusion equation. In our simulation, all three methods yield the same result, the effective diffusivity of an active colloid is the sum of the diffusivity of a passive colloid and ν(a)²τ/6. When the active colloids are confined by a harmonic well, they are trapped within a confinement length defined by the balance between the swimmer active force and restoring force of the well. When the confinement length is large compared to the run length, the stationary density profile follows the Boltzmann distribution. However, when the run length exceeds the confinement length, the density distribution is no longer described by Boltzmann distribution, instead we found a bimodal distribution.

Nonequilibrium Brownian motion beyond the effective temperature

The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einstein’s relation between mobility and diffusivity for Brownian motion. Several recent theories relax the hypothesis of thermal equilibrium resulting in at least two main scenarios. With well separated timescales, as in aging glassy systems, equilibrium Fluctuation-Dissipation Theorem applies at each scale with its own “effective” temperature. With mixed timescales, as for example in active or granular fluids or in turbulence, temperature is no more well-defined, the dynamical nature of fluctuations fully emerges and a Generalized Fluctuation-Dissipation Theorem (GFDT) applies. Here, we study experimentally the mixed timescale regime by studying fluctuations and linear response in the Brownian motion of a rotating intruder immersed in a vibro-fluidized granular medium. Increasing the packing fraction, the system is moved from a dilute single-timescale regime toward a denser multiple-timescale stage. Einstein’s relation holds in the former and is violated in the latter. The violation cannot be explained in terms of effective temperatures, while the GFDT is able to impute it to the emergence of a strong coupling between the intruder and the surrounding fluid. Direct experimental measurements confirm the development of spatial correlations in the system when the density is increased.

Chromosome positioning from activity-based segregation

Chromosomes within eukaryotic cell nuclei at interphase are not positioned at random, since gene-rich chromosomes are predominantly found towards the interior of the cell nucleus across a number of cell types. The physical mechanisms that could drive and maintain the spatial segregation of chromosomes based on gene density are unknown. Here, we identify a mechanism for such segregation, showing that the territorial organization of chromosomes, another central feature of nuclear organization, emerges naturally from our model. Our computer simulations indicate that gene density-dependent radial segregation of chromosomes arises as a robust consequence of differences in non-equilibrium activity across chromosomes. Arguing that such differences originate in the inhomogeneous distribution of ATP-dependent chromatin remodeling and transcription machinery on each chromosome, we show that a variety of non-random positional distributions emerge through the interplay of such activity, nuclear shape and specific interactions of chromosomes with the nuclear envelope. Results from our model are in reasonable agreement with experimental data and we make a number of predictions that can be tested in experiments.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

Driving kinetically constrained models into nonequilibrium steady states: Structural and slow transport properties

Complex fluids in shear flow and biased dynamics in crowded environments exhibit counterintuitive features which are difficult to address both at a theoretical level and by molecular dynamic simulations. To understand some of these features we study a schematic model of a highly viscous liquid, the two-dimensional Kob-Andersen kinetically constrained model, driven into nonequilibrium steady states by a uniform non-Hamiltonian force. We present a detailed numerical analysis of the microscopic behavior of the model, including transversal and longitudinal spatial correlations and dynamic heterogeneities. In particular, we show that at high particle density the transition from positive to negative resistance regimes in the current vs field relation can be explained via the emergence of nontrivial structures that intermittently trap the particles and slow down the dynamics. We relate such spatial structures to the current vs field relation in the different transport regimes.

Soft matter approaches to food structuring

We give an overview of the many opportunities that arise from approaching food structuring from the perspective of soft matter physics. This branch of physics employs concepts that build upon the seminal work of van der Waals, such as free volume, the mean field, and effective temperatures. All these concepts aid scientists in understanding and controlling the thermodynamics and (slow) dynamics of structured foods. We discuss the use of these concepts in four topics, which will also be addressed in a forthcoming Faraday Discussion on food structuring.

Crystallization in a dense suspension of self-propelled particles

Using Brownian dynamics computer simulations, we show that a two-dimensional suspension of self-propelled ("active") colloidal particles crystallizes at sufficiently high densities. Compared to the equilibrium freezing of passive particles, the freezing density is both significantly shifted and depends on the structural or dynamical criterion employed. In nonequilibrium the transition is accompanied by pronounced structural heterogeneities. This leads to a transition region between liquid and solid in which the suspension is globally ordered but unordered liquidlike "bubbles" still persist.

Communication: Effective temperature and glassy dynamics of active matter

A systematic expansion of the many-body master equation for active matter, in which motors power configurational changes as in the cytoskeleton, is shown to yield a description of the steady state and responses in terms of an effective temperature. The effective temperature depends on the susceptibility of the motors and a Peclet number which measures their strength relative to thermal Brownian diffusion. The analytic prediction is shown to agree with previous numerical simulations and experiments. The mapping also establishes a description of aging in active matter that is also kinetically jammed.

Effective interactions between colloidal particles suspended in a bath of swimming cells

The dynamics of passive colloidal tracers in a bath of self-propelled particles is receiving a lot of attention in the context of nonequilibrium statistical mechanics. Here we demonstrate that active baths are also capable of mediating effective interactions between suspended bodies. In particular we observe that a bath of swimming bacteria gives rise to a short range attraction similar to depletion forces in equilibrium colloidal suspensions. Using numerical simulations and experiments we show how the features of this interaction arise from the combination of nonequilibrium dynamics (peculiar of bacterial baths) and excluded volume effects.

Active colloidal suspensions exhibit polar order under gravity

Recently, the steady sedimentation profile of a dilute suspension of chemically powered colloids was studied experimentally [J. Palacci et al., Phys. Rev. Lett. 105, 088304 (2010)]. It was found that the sedimentation length increases quadratically with the swimming speed of the active Brownian particles. Here we investigate theoretically the sedimentation of self-propelled particles undergoing translational and rotational diffusion. We find that the measured increase of the sedimentation length is coupled to a partial alignment of the suspension with the mean swimming direction oriented against the gravitational field. We suggest realistic parameter values to observe this polar order. Furthermore, we find that the dynamics of the active suspension can be derived from a generalized free energy functional.

Perturbative path-integral study of active- and passive-tracer diffusion in fluctuating fields

We study the effective diffusion constant of a Brownian particle linearly coupled to a thermally fluctuating scalar field. We use a path-integral method to compute the effective diffusion coefficient perturbatively to lowest order in the coupling constant. This method can be applied to cases where the field is affected by the particle (an active tracer) and cases where the tracer is passive. Our results are applicable to a wide range of physical problems, from a protein diffusing in a membrane to the dispersion of a passive tracer in a random potential. In the case of passive diffusion in a scalar field, we show that the coupling to the field can, in some cases, speed up the diffusion corresponding to a form of stochastic resonance. Our results on passive diffusion are also confirmed via a perturbative calculation of the probability density function of the particle in a Fokker-Planck formulation of the problem. Numerical simulations on simplified systems corroborate our results.