Escape rate of active particles in the effective equilibrium approach

Confined active particles with spatially dependent Lorentz force: An odd twist to the "best Fokker-Planck approximation"

We derive a version of the so-called "best Fokker-Planck approximation" (BFPA) to describe the spatial properties of interacting active Ornstein-Uhlenbeck particles in arbitrary spatial dimensions. In doing so, we also take into account the odd-diffusive contribution of the Lorentz force acting on a charged particle in a spatially dependent magnetic field, sticking to the overdamped limit. While the BFPA itself does not turn out to be widely useful, our general approach allows us to deduce an appropriate generalization of the Fox approximation, which we use to characterize the stationary behavior of a single active particle in an external potential by deriving analytic expressions for configurational probability distributions (or effective potentials). In agreement with computer simulations, our theory predicts that the Lorentz force reduces the effective attraction and thus the probability to find an active particle in the vicinity of a repulsive wall. Even for an inhomogeneous magnetic field, our theoretical findings provide useful qualitative insights, specifically regarding the location of accumulation regions.

Run-and-tumble particle in one-dimensional potentials: Mean first-passage time and applications

We study a one-dimensional run-and-tumble particle (RTP), which is a prototypical model for active systems, moving within an arbitrary external potential. Using backward Fokker-Planck equations, we derive the differential equation satisfied by its mean first-passage time (MFPT) to an absorbing target, which, without any loss of generality, is placed at the origin. Depending on the shape of the potential, we identify four distinct "phases," with a corresponding expression for the MFPT in every case, which we derive explicitly. To illustrate these general expressions, we derive explicit formulas for two specific cases which we study in detail: a double-well potential and a logarithmic potential. We then present different applications of these general formulas to (1) the generalization of the Kramers escape law for an RTP in the presence of a potential barrier, (2) the "trapping" time of an RTP moving in a harmonic well, and (3) characterizing the efficiency of the optimal search strategy of an RTP subjected to stochastic resetting. Our results reveal that the MFPT of an RTP in an external potential exhibits a far more complex and, at times, counterintuitive behavior compared to that of a passive particle (e.g., Brownian) in the same potential.

Double Mpemba effect in the cooling of trapped colloids

The Mpemba effect describes the phenomenon that a system at hot initial temperature cools faster than at an initial warm temperature in the same environment. Such an anomalous cooling has recently been predicted and realized for trapped colloids. Here, we investigate the freezing behavior of a passive colloidal particle by employing numerical Brownian dynamics simulations and theoretical calculations with a model that can be directly tested in experiments. During the cooling process, the colloidal particle exhibits multiple non-monotonic regimes in cooling rates, with the cooling time decreasing twice as a function of the initial temperature-an unexpected phenomenon we refer to as the Double Mpemba effect. In addition, we demonstrate that both the Mpemba and Double Mpemba effects can be predicted by various machine-learning methods, which expedite the analysis of complex, computationally intensive systems.

Barrier crossing in a viscoelastic medium under active noise: Predictions of Kramers' flux-over-population method

The biochemical activity inside a cell has recently been suggested to act as a source of hydrodynamic fluctuations that can speed up or slow down enzyme catalysis [Tripathi et al., Commun. Phys. 5, 101 (2022).] The idea has been tested against and largely corroborated by simulations of activated barrier crossing in a simple fluid in the presence of thermal and athermal noise. The present paper attempts a wholly analytic solution to the same noise-driven barrier crossing problem but generalizes it to include viscoelastic memory effects of the kind likely to be present in cellular interiors. A calculation of the model's barrier crossing rate, using Kramers' flux-over-population formalism, reveals that in relation to the case where athermal noise is absent, athermal noise always accelerates barrier crossing, though the extent of enhancement depends on the duration τ0 over which the noise acts. More importantly, there exists a critical τ0-determined by the properties of the medium-at which Kramers' theory breaks down and, on approach to which, the rate grows significantly. The possibility of such a giant enhancement is potentially open to experimental validation using optically trapped nanoparticles in viscoelastic media that are acted on by externally imposed colored noise.

Time-delayed Duffing oscillator in an active bath

During recent decades active particles have attracted an incipient attention as they have been observed in a broad class of scenarios, ranging from bacterial suspension in living systems to artificial swimmers in nonequilibirum systems. The main feature of these particles is that they are able to gain kinetic energy from the environment, which is widely modeled by a stochastic process due to both (Gaussian) white and Ornstein-Uhlenbeck noises. In the present work, we study the nonlinear dynamics of the forced, time-delayed Duffing oscillator subject to these noises, paying special attention to their impact upon the maximum oscillations amplitude and characteristic frequency of the steady state for different values of the time delay and the driving force. Overall, our results indicate that the role of the time delay is substantially modified with respect to the situation without noise. For instance, we show that the oscillations amplitude grows with increasing noise strength when the time delay acts as a damping term in absence of noise, whereas the trajectories eventually become aperiodic when the oscillations are sustained by the time delay. In short, the interplay among the noises, forcing, and time delay gives rise to a rich dynamics: a regular and periodic motion is destroyed or restored owing to the competition between the noise and the driving force depending on time delay values, whereas an erratic motion insensitive to the driving force emerges when the time delay makes the motion aperiodic. Interestingly, we also show that, for a sufficient noise strength and forcing amplitude, an approximately periodic interwell motion is promoted by means of stochastic resonance.

Noise-induced escape of a self-propelled particle from metastable orbits

Active particles, like motile microorganisms and active colloids, are often found in confined environments where they can be arrested in a persistent orbital motion. Here, we investigate noise-induced switching between different coexisting orbits of a confined active particle as a stochastic escape problem. We show that, in the low-noise regime, this problem can be formulated as a least-action principle, which amounts to finding the most probable escape path from an orbit to the basin of attraction of another coexisting orbit. The corresponding action integral coincides with the activation energy, a quantity readily accessible in experiments and simulations via escape rate data. To illustrate how this approach can be used to tackle specific problems, we calculate optimum escape paths and activation energies for noise-induced transitions between clockwise and counterclockwise circular orbits of an active particle in radially symmetric confinement. We also investigated transitions between orbits of different topologies (ovals and lemniscates) coexisting in elliptic confinement. In all worked examples, the calculated optimum paths and minimum actions are in excellent agreement with mean-escape-time data obtained from direct numerical integration of the Langevin equations.

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.

Tailoring the escape rate of a Brownian particle by combining a vortex flow with a magnetic field

The probability per unit time for a thermally activated Brownian particle to escape over a potential well is, in general, well-described by Kramers's theory. Kramers showed that the escape time decreases exponentially with increasing barrier height. The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an external magnetic field. This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers's escape rate without any impact on the barrier-height-dependent exponent. Here, we show that the barrier height can be effectively changed when the charged particle is subjected to a vortex flow. While the vortex alone does not affect the mean escape time of the particle, when combined with a magnetic field, it effectively pushes the fluctuating particle either radially outside or inside depending on its sign relative to that of the magnetic field. In particular, the effective potential over which the particle escapes can be changed to a flat, a stable, and an unstable potential by tuning the signs and magnitudes of the vortex and the applied magnetic field. Notably, the last case corresponds to enhanced escape dynamics.

Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves

We report a systematic study of the dynamics of self-propelled particles (SPPs) over a one-dimensional periodic potential landscape U_{0}(x), which is fabricated on a microgroove-patterned polydimethylsiloxane (PDMS) substrate. From the measured nonequilibrium probability density function P(x;F_{0}) of the SPPs, we find that the escape dynamics of the slow rotating SPPs across the potential landscape can be described by an effective potential U_{eff}(x;F_{0}), once the self-propulsion force F_{0} is included into the potential under the fixed angle approximation. This work demonstrates that the parallel microgrooves provide a versatile platform for a quantitative understanding of the interplay among the self-propulsion force F_{0}, spatial confinement by U_{0}(x), and thermal noise, as well as its effects on activity-assisted escape dynamics and transport of the SPPs.

Anomalous Cooling and Overcooling of Active Colloids

The phenomenon that a system at a hot temperature cools faster than at a warm temperature, referred to as the Mpemba effect, has recently been realized for trapped colloids. Here, we investigate the cooling and heating process of a self-propelled active colloid using numerical simulations and theoretical calculations with a model that can be directly tested in experiments. Upon cooling, activity induces a Mpemba effect and the active particle transiently escapes an effective temperature description. At the end of the cooling process the notion of temperature is recovered and the system can exhibit even smaller temperatures than its final temperature, a surprising phenomenon which we refer to as activity-induced overcooling.

Active colloidal molecules in activity gradients

We consider a rigid assembly of two active Brownian particles, forming an active colloidal dimer, in a gradient of activity. We show analytically that depending on the relative orientation of the two particles the active dimer accumulates in regions of either high or low activity, corresponding to, respectively, chemotaxis and antichemotaxis. Certain active dimers show both chemotactic and antichemotactic behavior, depending on the strength of the activity. Our coarse-grained Fokker-Planck approach yields an effective potential, which we use to construct a nonequilibrium phase diagram that classifies the dimers according to their tactic behavior. Moreover, we show that for certain dimers a higher persistence of the motion is achieved similar to the effect of a steering wheel in macroscopic devices. This work could be useful for designing autonomous active colloidal structures which adjust their motion depending on the local activity gradients.

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.

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.

Active Ornstein-Uhlenbeck model for self-propelled particles with inertia

Self-propelled particles, which convert energy into mechanical motion, exhibit inertia if they have a macroscopic size or move inside a gaseous medium, in contrast to micron-sized overdamped particles immersed in a viscous fluid. Here we study an extension of the active Ornstein-Uhlenbeck model, in which self-propulsion is described by colored noise, to access these inertial effects. We summarize and discuss analytical solutions of the particle's mean-squared displacement and velocity autocorrelation function for several settings ranging from a free particle to various external influences, like a linear or harmonic potential and coupling to another particle via a harmonic spring. Taking into account the particular role of the initial particle velocity in a nonstationary setup, we observe all dynamical exponents between zero and four. After the typical inertial time, determined by the particle's mass, the results inherently revert to the behavior of an overdamped particle with the exception of the harmonically confined systems, in which the overall displacement is enhanced by inertia. We further consider an underdamped model for an active particle with a time-dependent mass, which critically affects the displacement in the intermediate time-regime. Most strikingly, for a sufficiently large rate of mass accumulation, the particle's motion is completely governed by inertial effects as it remains superdiffusive for all times.

Dissipation bounds the amplification of transition rates far from equilibrium

Complex systems can convert energy imparted by nonequilibrium forces to regulate how quickly they transition between long-lived states. While such behavior is ubiquitous in natural and synthetic systems, currently there is no general framework to relate the enhancement of a transition rate to the energy dissipated or to bound the enhancement achievable for a given energy expenditure. We employ recent advances in stochastic thermodynamics to build such a framework, which can be used to gain mechanistic insight into transitions far from equilibrium. We show that under general conditions, there is a basic speed limit relating the typical excess heat dissipated throughout a transition and the rate amplification achievable. We illustrate this tradeoff in canonical examples of diffusive barrier crossings in systems driven with autonomous and deterministic external forcing protocols. In both cases, we find that our speed limit tightly constrains the rate enhancement.

Solution to the Kramers barrier crossing problem caused by two noises: Thermal noise and Poisson white noise

We consider the escape of a particle trapped in a metastable potential well and acted upon by two noises. One of the noises is thermal and the other is Poisson white noise, which is non-Gaussian. Using path integral techniques, we find an analytic solution to this generalization of the classic Kramers barrier crossing problem. Using the "barrier climbing" path, we calculate the activation exponent. We also derive an approximate expression for the prefactor. The calculated results are compared with the simulations, and a good agreement between the two is found. Our results show that, unlike in the case of thermal noise, the rate depends not just on the barrier height but also on the shape of the whole barrier. A comparison between the simulations and the theory also shows that a better approximation for the prefactor is needed for agreement for all values of the parameters.

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.

Escape of a passive particle from an activity-induced energy landscape: emergence of slow and fast effective diffusion

Spontaneous persistent motions driven by active processes play a central role in maintaining living cells far from equilibrium. In the majority of research studies, the steady state dynamics of an active system has been described in terms of an effective temperature. By contrast, we have examined a prototype model for diffusion in an activity-induced rugged energy landscape to describe the slow dynamics of a tagged particle in a dense active environment. The expression for the mean escape time from the activity-induced rugged energy landscape holds only in the limit of low activity and the mean escape time from the rugged energy landscape increases with activity. The precise form of the active correlation will determine whether the mean escape time will depend on the persistence time or not. The activity-induced rugged energy landscape approach also allows an estimate of the non-equilibrium effective diffusivity characterizing the slow diffusive motion of the tagged particle due to activity. On the other hand, in a dilute environment, high activity augments the diffusion of the tagged particle. The enhanced diffusion can be attributed to an effective temperature higher than the ambient temperature and this is used to calculate the Kramers' mean escape time, which decreases with activity. Our results have direct relevance to recent experiments on tagged particle diffusion in condensed phases.

Chemokinetic Scattering, Trapping, and Avoidance of Active Brownian Particles

We present a theory of chemokinetic search agents that regulate directional fluctuations according to distance from a target. A dynamic scattering effect reduces the probability to penetrate regions with high fluctuations and thus reduces search success for agents that respond instantaneously to positional cues. In contrast, agents with internal states that initially suppress chemokinesis can exploit scattering to increase their probability to find the target. Using matched asymptotics between the case of diffusive and ballistic search, we obtain analytic results beyond Fox colored noise approximation.

Active Trap Model

Motivated by the dynamics of particles embedded in active gels, both in vitro and inside the cytoskeleton of living cells, we study an active generalization of the classical trap model. We demonstrate that activity leads to dramatic modifications in the diffusion compared to the thermal case: the mean square displacement becomes subdiffusive, spreading as a power law in time, when the trap depth distribution is a Gaussian and is slower than any power law when it is drawn from an exponential distribution. The results are derived for a simple, exactly solvable, case of harmonic traps. We then argue that the results are robust for more realistic trap shapes when the activity is strong.

Nondiffusive fluxes in a Brownian system with Lorentz force

The Fokker-Planck equation provides a complete statistical description of a particle undergoing random motion in a solvent. In the presence of Lorentz force due to an external magnetic field, the Fokker-Planck equation picks up a tensorial coefficient, which reflects the anisotropy of the particle's motion. This tensor, however, cannot be interpreted as a diffusion tensor; there are antisymmetric terms which give rise to fluxes perpendicular to the density gradients. Here, we show that for an inhomogeneous magnetic field these nondiffusive fluxes have finite divergence and therefore affect the density evolution of the system. Only in the special cases of a uniform magnetic field or carefully chosen initial condition with the same full rotational symmetry as the magnetic field can these fluxes be ignored in the density evolution.

Escape rate of transiently active Brownian particle in one dimension

Activity significantly enhances the escape rate of a Brownian particle over a potential barrier. Whereas constant activity has been extensively studied in the past, little is known about the effect of time-dependent activity on the escape rate of the particle. In this paper, we study the escape problem for a Brownian particle that is transiently active; the activity decreases rapidly during the escape process. Using the effective equilibrium approach, we analytically calculate the escape rate under the assumption that the particle is either completely passive or fully active when crossing the barrier. We perform numerical simulations of the escape process in one dimension and find good agreement with the theoretical predictions.

Diffusion in a bistable system: The eigenvalue spectrum of the Fokker-Planck operator and Kramers' reaction rate theory

The time-dependent solution of the Fokker-Planck equation with bistable potentials is considered in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck operator. The Fokker-Planck equation is the high friction limit of the corresponding Kramers' equation. Two different potentials are considered defined with a constant diffusion coefficient, ε, and position-dependent drift coefficients. The smallest nonzero eigenvalue of the Fokker-Planck operator, λ_{1}, provides the long-time rate coefficient for the transformation of the different species in the two stable states. A novel pseudospectral method with nonclassical polynomials is applied to this class of systems. The convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator versus the number of basis functions is studied and compared with previous results. The results are consistent with Kramers' theory, and a linear relationship between lnλ_{1} and 1/ε for sufficiently small ε values is verified. A comparison with analytic approximations to λ_{1} is provided.

Activity induced delocalization and freezing in self-propelled systems

We study a system of interacting active particles, propelled by colored noises, characterized by an activity time τ, and confined by a single-well anharmonic potential. We assume pair-wise repulsive forces among particles, modelling the steric interactions among microswimmers. This system has been experimentally studied in the case of a dilute suspension of Janus particles confined through acoustic traps. We observe that already in the dilute regime - when inter-particle interactions are negligible - increasing the persistent time, τ, pushes the particles away from the potential minimum, until a saturation distance is reached. We compute the phase diagram (activity versus interaction length), showing that the interaction does not suppress this delocalization phenomenon but induces a liquid- or solid-like structure in the densest regions. Interestingly a reentrant behavior is observed: a first increase of τ from small values acts as an effective warming, favouring fluidization; at higher values, when the delocalization occurs, a further increase of τ induces freezing inside the densest regions. An approximate analytical scheme gives fair predictions for the density profiles in the weakly interacting case. The analysis of non-equilibrium heat fluxes reveals that in the region of largest particle concentration equilibrium is restored in several aspects.

Thermal decay of a metastable state: Influence of rescattering on the quasistationary dynamical rate

We study the effect of backscattering of the Brownian particles as they escape out of a metastable state overcoming the potential barrier. For this aim, we model this process numerically using the Langevin equations. This modeling is performed for the wide range of the friction constant covering both the energy and spatial diffusion regimes. It is shown how the influence of the descent stage on the quasistationary decay rate gradually disappears as the friction constant decreases. It is found that, in the energy diffusion regime, the rescattering absents and the descent stage does not influence the decay rate. As the value of friction increases, the descent alters the value of the rate by more than 50% for different values of thermal energy and different shapes of the potential. To study the influence of the backscattering on the decay rate, four potentials have been considered which coincide near the potential well and the barrier but differ beyond the barrier. It is shown that the potential for which the well and the barrier are described by two smoothly joined parabolas ("the parabolic potential") plays a role of a dividing range for the mutual layout of the quasistationary dynamical rate and the widely used in the literature Kramers rate. Namely, for the potentials with steeper tails, the Kramers rate R_{KM} underestimates the true quasistationary dynamical rate R_{D}, whereas for the less steep tails the opposite holds (inversion of R_{D}/R_{KM}). It is demonstrated that the mutual layout of the values of R_{D} for different potentials is explained by the rescattering of the particles from the potential tail.

Pseudochemotaxis in inhomogeneous active Brownian systems

We study dynamical properties of confined, self-propelled Brownian particles in an inhomogeneous activity profile. Using Brownian dynamics simulations, we calculate the probability to reach a fixed target and the mean first passage time to the target of an active particle. We show that both these quantities are strongly influenced by the inhomogeneous activity. When the activity is distributed such that high-activity zone is located between the target and the starting location, the target finding probability is increased and the passage time is decreased in comparison to a uniformly active system. Moreover, for a continuously distributed profile, the activity gradient results in a drift of active particle up the gradient bearing resemblance to chemotaxis. Integrating out the orientational degrees of freedom, we derive an approximate Fokker-Planck equation and show that the theoretical predictions are in very good agreement with the Brownian dynamics simulations.

Effective equilibrium states in mixtures of active particles driven by colored noise

We consider the steady-state behavior of pairs of active particles having different persistence times and diffusivities. To this purpose we employ the active Ornstein-Uhlenbeck model, where the particles are driven by colored noises with exponential correlation functions whose intensities and correlation times vary from species to species. By extending Fox's theory to many components, we derive by functional calculus an approximate Fokker-Planck equation for the configurational distribution function of the system. After illustrating the predicted distribution in the solvable case of two particles interacting via a harmonic potential, we consider systems of particles repelling through inverse power-law potentials. We compare the analytic predictions to computer simulations for such soft-repulsive interactions in one dimension and show that at linear order in the persistence times the theory is satisfactory. This work provides the toolbox to qualitatively describe many-body phenomena, such as demixing and depletion, by means of effective pair potentials.

Mode coupling theory for nonequilibrium glassy dynamics of thermal self-propelled particles

We present a mode coupling theory study for the relaxation and glassy dynamics of a system of strongly interacting self-propelled particles, wherein the self-propulsion force is described by Ornstein-Uhlenbeck colored noise and thermal noises are included. Our starting point is an effective Smoluchowski equation governing the distribution function of particle positions, from which we derive a memory function equation for the time dependence of density fluctuations in nonequilibrium steady states. With the basic assumption of the absence of macroscopic currents and standard mode coupling approximation, we can obtain expressions for the irreducible memory function and other relevant dynamic terms, wherein the nonequilibrium character of the active system is manifested through an averaged diffusion coefficient D[combining macron] and a nontrivial structural function S₂(q) with q being the magnitude of wave vector q. D[combining macron] and S₂(q) enter the frequency term and the vertex term for the memory function, and thus influence both the short time and the long time dynamics of the system. With these equations obtained, we study the glassy dynamics of this thermal self-propelled particle system by investigating the Debye-Waller factor f_q and relaxation time τ_α as functions of the persistence time τ_p of self-propulsion, the single particle effective temperature T_eff as well as the number density ρ. Consequently, we find the critical density ρ_c for given τ_p shifts to larger values with increasing magnitude of propulsion force or effective temperature, in good accordance with previously reported simulation work. In addition, the theory facilitates us to study the critical effective temperature T for fixed ρ as well as its dependence on τ_p. We find that T increases with τ_p and in the limit τ_p → 0, it approaches the value for a simple passive Brownian system as expected. Our theory also well recovers the results for passive systems and can be easily extended to more complex systems such as active-passive mixtures.

Directional transport of colloids inside a bath of self-propelling walkers

We present a setup in which passive colloids inside a solvent are moved to the boundaries of the container. The directional transport is facilitated by self-propelling microparticles ("walkers") with an activity gradient, which reduces their propulsion in the vicinity of bounding walls. An attractive interaction leads to the adsorption of walkers onto the colloid-surfaces in regions of low walker activity. It is shown that the activity gradient generates a free energy gradient which in turn acts as a driving force on the passive colloids. We carry out molecular dynamics simulations and present approaches to a theoretical description of the involved processes. Although the simulation data are not reproduced on a fully quantitative level, their qualitative features are covered by the model. The effect described here may be applied to facilitate a directional transport of drugs or to eliminate pollutants.