We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent on the friction coefficient.
We demonstrate that there is macroscopic co-existence between regions with hexatic order and regions in the liquid/gas phase over a finite interval of packing fractions in active dumbbell systems with repulsive power-law interactions in two dimensions. In the passive limit this interval remains finite, similarly to what has been found in bidimensional systems of hard and soft disks. We did not find discontinuous behaviour upon increasing activity from the passive limit.
We study statistical properties of the process $Y(t)$ of a passive advection by quenched random layered flows in situations when the inter-layer transfer is governed by a fractional Brownian motion $X(t)$ with the Hurst index $H \in (0,1)$. We show that the disorder-averaged mean-squared displacement of the passive advection grows in the large time $t$ limit in proportion to $t^{2 - H}$, which defines a family of anomalous super-diffusions. We evaluate the disorder-averaged Wigner-Ville spectrum of the advection process $Y(t)$ and demonstrate that it has a rather unusual power-law form $1/f^{3…
Suspensions of purely repulsive but self-propelled Brownian particles might undergo phase separation, a phenomenon that strongly resembles the phase separation of passive particles with attractions. Here we employ computer simulations to study the nucleation kinetics and the microscopic pathway active Brownian disks take in two dimensions when quenched from the homogeneous suspension to propulsion speeds beyond the binodal. We find the same qualitative behavior for the nucleation rate as a function of density as for a passive suspension undergoing liquid-vapor separation, suggesting that the s…
Active particles may happen to be confined in channels so narrow that they cannot overtake each other (Single File conditions). This interesting situation reveals nontrivial physical features as a consequence of the strong inter-particle correlations developed in collective rearrangements. We consider a minimal model for active Brownian particles with the aim of studying the modifications introduced by activity with respect to the classical (passive) Single File picture. Depending on whether their motion is dominated by translational or rotational diffusion, we find that active Brownian partic…
We study the two-dimensional motion of an active Brownian particle of speed $v_0$, with intermittent directional reversals in the presence of a harmonic trap of strength $μ$. The presence of the trap ensures that the position of the particle eventually reaches a steady state where it is bounded within a circular region of radius $v_0/μ$, centered at the minimum of the trap. Due to the interplay between the rotational diffusion constant $D_R$, reversal rate $γ$, and the trap strength $μ$, the steady state distribution shows four different types of shapes, which we refer to as active-I & II, and…
We study the effects of stochastic birth and death processes on the structural phases of systems of active and passive Brownian particles subject to volume exclusion. The total number of particles in the system is a fluctuating quantity, determined by the birth and death parameters, and on the activity of the particles. As the birth and death parameters are varied we find liquid, hexatic and solid phases. For passive particles these phases are found to be spatially homogeneous. For active particles motility-induced phase separation (co-existing hexatic and liquid phases) occurs for large activ…
These notes focus on the description of the phases of matter in two dimensions. Firstly, we present a brief discussion of the phase diagrams of bidimensional interacting passive systems, and their numerical and experimental measurements. The presentation will be short and schematic. We will complement these notes with a rather complete bibliography that should guide the students in their study of the development of this very rich subject over the last century. Secondly, we summarise very recent results on the phase diagrams of active Brownian disks and active dumbbell systems in two dimensions…
The equilibrium properties of a system of passive diffusing particles in an external magnetic field are unaffected by the Lorentz force. In contrast, active Brownian particles exhibit steady-state phenomena that depend on both the strength and the polarity of the applied magnetic field. The intriguing effects of the Lorentz force, however, can only be observed when out-of-equilibrium density gradients are maintained in the system. To this end, we use the method of stochastic resetting on active Brownian particles in two dimensions by resetting them to the line $x=0$ at a constant rate and peri…
We perform an extensive analysis of passive as well as active micro-heat engines with different single-particle stochastic models. Using stochastic thermodynamics we calculate thermodynamic work, heat, entropy production and efficiency of passive and active Brownian heat engines analytically as well as numerically and compare them. We run the heat engines with a protocol for which the average thermodynamic quantities are calculated exactly for an arbitrary cycle time. We also discuss about the group of protocols for which exact non-quasistatic calculations can be done, completely in the passiv…
By collecting from literature data the experimental evidences of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live E. coli cells, we get the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Krätzel function and its Fokker-Planck equation be a fractional diffusion equation in the Erdélyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.
One of the most remarkable observations in dense active matter systems is the appearance of long-range velocity correlations without any explicit aligning interaction (of e.g.\ Vicsek type). Here we show that this kind of long range velocity correlation can also be generated in a dense athermal passive system by the inclusion of a very small fraction of active Brownian particles. We develop a continuum theory to explain the emergence of velocity correlations generated via such active dopants. We validate the predictions for the effects of magnitude and persistence time of the active force and …
We find an exact series solution for the steady-state probability distribution of a harmonically trapped active Brownian particle in two dimensions, in the presence of translational diffusion. This series solution allows us to efficiently explore the behavior of the system in different parameter regimes. Identifying "active" and "passive" regimes, we predict a surprising re-entrant active-to-passive transition with increasing trap stiffness. Our numerical simulations validate this finding. We discuss various interesting limiting cases wherein closed form expressions for the distributions can b…
We have performed a case study investigating a recently proposed scheme to obtain an effective pair potential for active Brownian particles [Farage et al., Phys. Rev. E 91, 042310 (2015)]. Applying this scheme to the Lennard-Jones potential, numerical simulations of active Brownian particles are compared to simulations of passive Brownian particles interacting by the effective pair potential. Analyzing the static pair correlations, our results indicate a limited range of activity parameters (speed and orientational correlation time) for which we obtain quantitative, or even qualitative, agreem…
The kinetic separation of repulsive active Brownian particles into a dense and a dilute phase is analyzed using a systematic coarse-graining strategy. We derive an effective Cahn-Hilliard equation on large length and time scales, which implies that the separation process can be mapped onto that of passive particles. A lower density threshold for clustering is found, and using our approach we demonstrate that clustering first proceeds via a hysteretic nucleation scenario and above a higher threshold changes into a spinodal-like instability. Our results are in agreement with particle-resolved co…
We investigate the mean first passage time of an active Brownian particle in one dimension using numerical simulations. The activity in one dimension is modeled as a two state model; the particle moves with a constant propulsion strength but its orientation switches from one state to other as in a random telegraphic process. We study the influence of a finite resetting rate $r$ on the mean first passage time to a fixed target of a single free Active Brownian Particle and map this result using an effective diffusion process. As in the case of a passive Brownian particle, we can find an optimal …
We investigate the phase behavior and kinetics of a monodisperse mixture of active (\textit{i.e.}, self-propelled) and passive isometric Brownian particles through Brownian dynamics simulations and theory. As in a purely active system, motility of the active component triggers phase separation into a dense and a dilute phase; in the dense phase we further find active-passive segregation, with "rafts" of passive particles in a "sea" of active particles. We find that phase separation from an initially disordered mixture can occur with as little as 15 percent of the particles being active. Finall…
We model a binary mixture of passive and active Brownian particles in two dimensions using the effective interaction between passive particles in the active bath. The activity of active particles and the size ratio of two types of particles are two control parameters in the system. The effective interaction is calculated from the average force on two particles generated by the active particles. The effective interaction can be attractive or repulsive, depending on the system parameters. The passive particles form four distinct structural orders for different system parameters viz; disorder (D)…
A particle that is immersed in a uniform temperature bath performs Brownian diffusion, as discussed by Einstein. But Sinai has realized that in a "random environment" the diffusion is suppressed. Follow-up works have pointed out that in the presence of bias $f$ there are delocalization and sliding transitions, with threshold value $f_c$ that depends on the disorder strength. We discuss in a critical way the emergence of Sinai physics for both passive and active Brownian particles. Tight-binding and Fokker-Planck versions of the model are addressed on equal footing. We assume that the transitio…
We derive from first principles the mechanical pressure $P$, defined as the force per unit area on a bounding wall, in a system of spherical, overdamped, active Brownian particles at density $ρ$. Our exact result relates $P$, in closed form, to bulk correlators and shows that (i) $P(ρ)$ is a state function, independent of the particle-wall interaction; (ii) interactions contribute two terms to $P$, one encoding the slow-down that drives motility-induced phase separation, and the other a direct contribution well known for passive systems; (iii) $P(ρ)$ is equal in coexisting phases. We discuss t…
We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion as exhibited, for example, by the dynamics of a chemically-active Janus colloid. We evaluate the standardly-defined spectral density, i.e. the STSD averaged over a statistical ensemble of trajectories in the limit of an infinitely long observation time $T$, and also go beyond the standard analysis by considering the coefficient of variation $γ$ of the distribution of the STSD. Moreover, we analyse the finite-$T$ behaviour of the STSD and $γ$, determine the cross-correlations between spatial compone…
We consider a model of a two-dimensional molecular machine - called Brownian gyrator - that consists of two coordinates coupled to each other and to separate heat baths at temperatures respectively $T_x$ and $T_y$. We consider the limit in which one component is passive, because its bath is "cold", $T_x \to 0$, while the second is in contact with a "hot" bath, $T_y > 0$, hence it entrains the passive component in a stochastic motion. We derive an asymmetry relation as a function of time, from which time dependent effective temperatures can be obtained for both components. We find that the effe…
Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Phys. Rev. Lett. 112, 218304 (2014)]. Here we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus…
We study the Brownian motion of particles trapped by optical tweezers inside a colloidal glass (Laponite) during the sol-gel transition. We use two methods based on passive rheology to extract the effective temperature from the fluctuations of the Brownian particles. All of them give a temperature that, within experimental errors, is equal to the heat bath temperature. Several interesting features concerning the statistical properties and the long time correlations of the particles are observed during the transition.
We consider a model of active Brownian particles with velocity-alignment in two spatial dimensions with passive and active fluctuations. Hereby, active fluctuations refers to purely non-equilibrium stochastic forces correlated with the heading of an individual active particle. In the simplest case studied here, they are assumed as independent stochastic forces parallel (speed noise) and perpendicular (angular noise) to the velocity of the particle. On the other hand, passive fluctuations are defined by a noise vector independent of the direction of motion of a particle, and may account for exa…
We study by simulation a mixture of active (run-and-tumble) and passive (Brownian) particles with repulsive exclusion interactions in one dimension, subject to a ratchet (smoothed sawtooth) potential. Such a potential is known to rectify active particles at one-body level, creating a net current in the `easy direction'. This is the direction in which one encounters the lower maximum force en route to the top of a potential barrier. The exclusion constraint results in single-file motion, so the mean velocities of active and passive particles are identical; we study the effects of activity level…
We investigate a model for a Stirling-like engine consisting of a passive Brownian particle confined by a harmonic potential and interacting with a suspension of active Brownian particles that self-propel in a viscous solvent, which cyclically operates under isothermal conditions by means of temporal variations of the trap stiffness and the self-propulsion speed of the active particles. We derive an effective stochastic equation of motion of the trapped Brownian particle, which includes a friction memory kernel as well as thermal and active fluctuating forces due to its coupling with the activ…
Moving against the odds with a cyclic active Brownian ratchet: current enhancement and reversal
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Institute for the Dynamics of Complex Systems, University of Göttingen（哥廷根大学复杂系统动力学研究所）；Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS（蒙彼利埃大学查尔斯·库隆实验室（L2C），法国国家科学研究中心）
We study a Brownian particle passively driven by a field obeying the noisy Burgers equation. We demonstrate that the system exhibits replica symmetry breaking in the path ensemble with the initial position of the particle being fixed. The key step of the proof is that the path ensemble with a modified boundary condition can be exactly mapped to the canonical ensemble of directed polymers.
By working in the small persistence time limit, we determine the steady-state distribution of an Active Ornstein Uhlenbeck Particle (AOUP) experiencing, in addition to self-propulsion, a Gaussian white noise modelling a bath at temperature T. This allows us to derive analytical formulas for three quantities: the spatial density of a confined particle, the current induced by an asymmetric periodic potential and the entropy production rate. These formulas disentangle the respective roles of the passive and active noises on the steady state of AOUPs, showing that signatures of non-equilibrium can…
It is well established that glassy materials can undergo aging, i.e., their properties gradually change over time. There is rapidly growing evidence that dense active and living systems also exhibit many features of glassy behavior, but it is still largely unknown how physical aging is manifested in such active glassy materials. Our goal is to explore whether active and passive thermal glasses age in fundamentally different ways. To address this, we numerically study the aging dynamics following a quench from high to low temperature for two-dimensional passive and active Brownian model glass-f…
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.
Active colloids exhibit persistent motion, which can lead to motility-induced phase separation (MIPS). However, there currently exists no microscopic theory to account for this phenomenon. We report a first-principles theory, free of fit parameters, for active spherical colloids, which shows explicitly how an effective many-body interaction potential is generated by activity and how this can rationalize MIPS. For a passively repulsive system the theory predicts phase separation and pair correlations in quantitative agreement with simulation. For an attractive system the theory shows that phase…
Understanding the out-of-equilibrium properties of noisy microscale systems and the extent to which they can be modulated externally, is a crucial scientific and technological challenge. It holds the promise to unlock disruptive new technologies ranging from targeted delivery of chemicals within the body to directed assembly of new materials. Here we focus on how active matter can be harnessed to transport passive microscopic systems in a statistically predictable way. Using a minimal active-passive system of weakly Brownian particles and swimming microalgae, we show that spatial confinement l…
Active matter systems are driven out of equilibrium by conversion of energy into directed motion locally on the level of the individual constituents. In the spirit of a minimal description, active matter is often modeled by so-called active Ornstein-Uhlenbeck particles (AOUPs), an extension of passive Brownian motion where activity is represented by an additional fluctuating non-equilibrium "force" with simple statistical properties (Ornstein-Uhlenbeck process). While in passive Brownian motion, entropy production along trajectories is well-known to relate to irreversibility in terms of the lo…
