Brownian motion of a self-propelled particle

Direct measurement of self-diffusiophoretic force generated by active colloids of different patch coverage using optical tweezers

HYPOTHESIS

Synthetic micro/nanomotors are gaining extensive attention for various biomedical applications (especially in drug delivery) due to their ability to mimic the motion of biological micro/nanoscale swimmers. The feasibility of these applications relies on tight control of propulsion speed, direction, and type of motion (translation, circular, etc.) along with the exerted self-propulsive force. We propose to exploit the variation of both self-propulsion speed and force of active colloids with different patch coverages (with and without supporting layer) for engineering diffusiophoretic micro/nanomotors.

EXPERIMENTS

The microswimmers were designed at various patch coverages (10°, 30°, and 90°) with (Ti/Pt) and without (Pt) an adhesion layer for the catalytic patch through glancing angle metal deposition (GLAD) technique. Mean-square displacement (MSD) analysis was performed to obtain the self-propulsion parameters like speed and angular speed. Using optical tweezers (OT), the self-propulsive force was measured from the force power spectral density.

FINDINGS

The findings of our experiments suggest the non-requirement of any adhesion layer preceding the catalyst deposition since the Pt 10° colloidal batch had the maximal self-propulsion speed (4.61±0.3μm/s) and force (345±57fN) for 5% w/v H2O2 fuel concentration. Moreover, the self-propulsion speed and force decreased with increasing patch size, contrary to theoretical estimates. Also, the self-propulsive force obtained from MSD is 2 to 4 times lower in magnitude than the OT based force values. We believe that the self-propelling motion of the micromotors is possibly hindered due to interactions with the surface of the quartz cuvette during the optical microscopic analysis. Further, the MSD is limited to the self-propulsive motion in two dimensions. On the other hand, OT based force measurement involve trapping the particles in the bulk of the solution entirely avoiding the particle-substrate interactions. Hence, OT based force measurements are better than the propulsion velocity based stokes drag force estimates. We believe that this study can lay the foundation in designing efficient micro/nanomotors for translational biomedical applications.

Active string fluids and gels formed by dipolar active Brownian particles in 3D

Self-propelled particles possessing permanent magnetic dipole moments occur naturally in magnetotactic bacteria and can be built into man-made systems such as active colloids or micro-robots. Yet, the interplay between self-propulsion and anisotropic dipole-dipole interactions on dynamic self-assembly in three dimensions (3D) remains poorly understood. We conduct Brownian dynamics simulations of active dipolar particles in 3D, focusing on the low-density regime, where dipolar hard spheres tend to form chain-like aggregates and percolated networks with increasing dipolar coupling strength. We find that strong active forces override dipolar attractions, effectively inhibiting chain-like aggregation and network formation. Conversely, activating particles with low to moderate forces results in a fluid composed of active chains and rings. At strong dipolar coupling strengths, this active fluid transitions into an active gel, consisting of a percolated network of active chains. Although the overall structure of the active gel remains interconnected, the network experiences more frequent configurational rearrangements due to the reduced bond lifetime of active dipolar particles. Consequently, particles exhibit enhanced translational and rotational diffusion within the active fluid of strings and active gels compared to their passive counterparts. We quantify the influence of activity on aggregate topology as they transition from branched structures to unconnected chains and rings. Our findings are summarized in a state diagram, delineating the impact of dipolar coupling strength and active force magnitude on the system.

Probing Asymmetric Interactions with Time-Separated Mutual Information: A Case Study Using Golden Shiners

Leader-follower modalities and other asymmetric interactions that drive the collective motion of organisms are often quantified using information theory metrics like transfer or causation entropy. These metrics are difficult to accurately evaluate without a much larger number of data than is typically available from a time series of animal trajectories collected in the field or from experiments. In this paper, we use a generalized leader-follower model to argue that the time-separated mutual information between two organism positions can serve as an alternative metric for capturing asymmetric correlations that is much less data intensive and more accurately estimated by popular k-nearest neighbor algorithms than transfer entropy. Our model predicts a local maximum of this mutual information at a time separation value corresponding to the fundamental reaction timescale of the follower organism. We confirm this prediction by analyzing time series trajectories recorded for a pair of golden shiner fish circling an annular tank.

Impact of non-reciprocal interactions on colloidal self-assembly with tunable anisotropy

Non-reciprocal (NR) effective interactions violating Newton's third law occur in many biological systems, but can also be engineered in synthetic, colloidal systems. Recent research has shown that such NR interactions can have tremendous effects on the overall collective behavior and pattern formation, but can also influence aggregation processes on the particle scale. Here, we focus on the impact of non-reciprocity on the self-assembly of a colloidal system (originally passive) with anisotropic interactions whose character is tunable by external fields. In the absence of non-reciprocity, that is, under equilibrium conditions, the colloids form square-like and hexagonal aggregates with extremely long lifetimes yet no large-scale phase separation [Kogler et al., Soft Matter 11, 7356 (2015)], indicating kinetic trapping. Here, we study, based on Brownian dynamics simulations in 2D, an NR version of this model consisting of two species with reciprocal isotropic, but NR anisotropic interactions. We find that NR induces an effective propulsion of particle pairs and small aggregates ("active colloidal molecules") forming at the initial stages of self-assembly, an indication of the NR-induced non-equilibrium. The shape and stability of these initial clusters strongly depend on the degree of anisotropy. At longer times, we find, for weak NR interactions, large (even system-spanning) clusters where single particles can escape and enter at the boundaries, in stark contrast to the small rigid aggregates appearing at the same time in the passive case. In this sense, weak NR shortcuts the aggregation. Increasing the degree of NR (and thus, propulsion), we even observe large-scale phase separation if the interactions are weakly anisotropic. In contrast, systems with strong NR and anisotropy remain essentially disordered. Overall, the NR interactions are shown to destabilize the rigid aggregates interrupting self-assembly and phase separation in the passive case, thereby helping the system to overcome kinetic barriers.

Anomalous diffusion of scaled Brownian tracers

A model for anomalous transport of tracer particles diffusing in complex media in two dimensions is proposed. The model takes into account the characteristics of persistent motion that an active bath transfers to the tracer; thus, the model proposed here extends active Brownian motion, for which the stochastic dynamics of the orientation of the propelling force is described by scaled Brownian motion (sBm), identified by time-dependent diffusivity of the form D_{β}∝t^{β-1}, β>0. If β≠1, sBm is highly nonstationary and suitable to describe such nonequilibrium dynamics induced by complex media. In this paper, we provide analytical calculations and computer simulations to show that genuine anomalous diffusion emerges in the long-time regime, with a time scaling of the mean-squared displacement t^{2-β}, while ballistic transport t^{2}, characteristic of persistent motion, is found in the short-time regime. We also analyze the time dependence of the kurtosis, and the intermediate scattering function of the position distribution, as well as the propulsion autocorrelation function, which defines the effective persistence time.

Extinction and coexistence in a binary mixture of proliferating motile disks

A binary mixture of two-different-size proliferating motile disks is studied. As growth is space limited, we focus on the conditions such that there is a coexistence of both large and small disks, or dominance of the larger disks. The study involves systematically varying some system parameters, such as diffusivities, growth rates, and self-propulsion velocities. In particular, we demonstrate that diffusing faster confers a competitive advantage, so that larger disks can in the long time coexist or even dominate the smaller ones. In the case of self-propelled disks, a coexistence regime is induced by the activity where the two types of disks show the same spatial distribution: both particles are phase separated or both are homogeneously distributed in the whole system.

Inertial active harmonic particle with memory induced spreading by viscoelastic suspension

We investigate the self-propulsion of an inertial active particle confined in a two-dimensional harmonic trap. The particle is suspended in a non-Newtonian or viscoelastic suspension with a friction kernel that decays exponentially with a time constant characterizing the memory timescale or transient elasticity of the medium. By solving the associated non-Markovian dynamics, we identify two regimes in parameter space distinguishing the oscillatory and non-oscillatory behavior of the particle motion. By simulating the particle trajectories and exactly calculating the steady-state probability distribution functions and mean square displacement; interestingly, we observe that with an increase in the memory time scale, the effective temperature of the environment increases. As a consequence, the particle becomes energetic and spread away from the center, covering larger space inside the confinement. On the other hand, with an increase in the duration of the activity, the particle becomes trapped by the harmonic confinement.

Emergent structures in active block copolymer composites

Block copolymer melts offer unique templates to control the position and alignment of nanoparticles due to their ability to self-assemble into periodic ordered structures. Active particles are shown to coassemble with block copolymers leading to emergent organized structures. The block copolymer acts as a soft template that can control the self-propulsion of active particles, both for interface-segregated and selective nanoparticles. At moderate activities, active particles can form organized structures such as polarized trains or rotating vortices. At high activity, the contrast in the polymeric and colloidal timescales can lead to particle swarms with distorted block copolymer morphology, due to the competition between polymeric self-assembly and active Brownian self-propulsion.

Stochastic motion under nonlinear friction representing shear thinning

We study stochastic motion under a nonlinear frictional force that levels off with increasing velocity. Specifically, our frictional force is of the so-called Coulomb-tanh type. At small speed, it increases approximately linearly with velocity, while at large speed, it approaches a constant magnitude, similarly to solid (dry, Coulomb) friction. In one spatial dimension, a formal analogy between the associated Fokker-Planck equation and the Schrödinger equation for a quantum mechanical oscillator in a nonharmonic Pöschl-Teller potential is revealed. Then, the stationary velocity statistics can be treated analytically. From such analytical considerations, we determine associated diffusion coefficients, which we confirm by agent-based simulations. Moreover, from such simulations and from numerically solving the associated Fokker-Planck equation, we find that the spatial distribution function, starting from an initial Gaussian shape, develops tails that appear exponential at intermediate timescales. At small magnitudes of stochastic driving, the velocity distribution resembles the case of linear friction, while at large magnitudes, it rather approaches the case of solid (dry, Coulomb) friction. The same is true for diffusion coefficients. In a certain sense thus interpolating between the two extreme cases of linear friction and solid (dry, Coulomb) friction, our approach should be useful to describe several cases of practical relevance. For instance, a reduced increase in friction with increasing relative speed is typical of shear-thinning behavior. Therefore, driven motion in shear-thinning environments is one specific example to which our description may be applied.

Mechanochemical active ratchet

Self-propelled nanoparticles moving through liquids offer the possibility of creating advanced applications where such nanoswimmers can operate as artificial molecular-sized motors. Achieving control over the motion of nanoswimmers is a crucial aspect for their reliable functioning. While the directionality of micron-sized swimmers can be controlled with great precision, steering nano-sized active particles poses a real challenge. One of the reasons is the existence of large fluctuations of active velocity at the nanoscale. Here, we describe a mechanism that, in the presence of a ratchet potential, transforms these fluctuations into a net current of active nanoparticles. We demonstrate the effect using a generic model of self-propulsion powered by chemical reactions. The net motion along the easy direction of the ratchet potential arises from the coupling of chemical and mechanical processes and is triggered by a constant, transverse to the ratchet, force. The current magnitude sensitively depends on the amplitude and the periodicity of the ratchet potential and the strength of the transverse force. Our results highlight the importance of thermodynamically consistent modeling of chemical reactions in active matter at the nanoscale and suggest new ways of controlling dynamics in such systems.

Effect of speed fluctuations on the collective dynamics of active disks

Numerical simulations are performed on the collective dynamics of active disks, whose self-propulsion speed (U) varies in time, and whose orientation evolves according to rotational Brownian motion. Two protocols for the evolution of speed are considered: (i) a deterministic one involving a periodic change in U at a frequency ω; and (ii) a stochastic one in which the speeds are drawn from a power-law distribution at time-intervals governed by a Poissonian process of rate β. In the first case, an increase in ω causes the disks to go from a clustered state to a homogeneous one through an apparent phase-transition, provided that the direction of self-propulsion is allowed to reverse. Similarly, in the second case, for a fixed value of β, the extent of cluster-breakup is larger when reversals in the self-propulsion direction are permitted. Motility-induced phase separation of the disks may therefore be avoided in active matter suspensions in which the constituents are allowed to reverse their self-propulsion direction, immaterial of the precise temporal nature of the reversal (deterministic or stochastic). Equally, our results demonstrate that phase separation could occur even in the absence of a time-averaged motility of an individual active agent, provided that the rate of direction reversals is smaller than the orientational diffusion rate.

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.

Inverted Sedimentation of Active Particles in Unbiased ac Fields

Gaining control over the motion of active particles is crucial for applications ranging from targeted cargo delivery to nanomedicine. While much progress has been made recently to control active motion based on external forces, flows, or gradients in concentration or light intensity, which all have a well-defined direction or bias, little is known about how to steer active particles in situations where no permanent bias can be realized. Here, we show that ac fields with a vanishing time average provide an alternative route to steering active particles. We exemplify this route for inertial active particles in a gravitational field, observing that a substantial fraction of them persistently travels in the upward direction upon switching on the ac field, resulting in an inverted sedimentation profile at the top wall of a confining container. Our results offer a generic control principle that could be used in the future to steer active motion, direct collective behaviors, and purify mixtures.

Most probable paths for active Ornstein-Uhlenbeck particles

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

Inertial and geometrical effects of self-propelled elliptical Brownian particles

Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.

Inertial active ratchet: Simulation versus theory

We present the inertial active dynamics of an Ornstein-Uhlenbeck particle in a piecewise sawtooth ratchet potential. Using the Langevin simulation and matrix continued fraction method (MCFM), the particle transport, steady-state diffusion, and coherence in transport are investigated in different parameter regimes of the model. Spatial asymmetry is found to be a key criterion for the possibility of directed transport in the ratchet. The MCFM results for net particle current of overdamped dynamics of the particle agree well with the simulation results. The simulated particle trajectories for the inertial dynamics and the corresponding position and velocity distribution functions reveal that the system passes through an activity-induced transition in the transport from the running phase to the locked phase of the dynamics. This is further corroborated by the mean square displacement (MSD) calculations, where the MSD gets suppressed with increase in the persistent duration of activity or self-propulsion in the medium and finally approaches zero for a very large value of self propulsion time. The nonmonotonic behavior of the particle current and Péclet number with self-propulsion time confirms that the particle transport and its coherence can be enhanced or reduced by fine tuning the persistent duration of activity. Moreover, for intermediate ranges of self-propulsion time as well as mass of the particle, even though the particle current shows a pronounced unusual maximum with mass, there is no enhancement in the Péclet number, instead the Péclet number decreases with mass, confirming the degradation of coherence in transport.

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.

Exploiting compositional disorder in collectives of light-driven circle walkers

Emergent behavior in collectives of "robotic" units with limited capabilities that is robust and programmable is a promising route to perform tasks on the micro and nanoscale that are otherwise difficult to realize. However, a comprehensive theoretical understanding of the physical principles, in particular steric interactions in crowded environments, is still largely missing. Here, we study simple light-driven walkers propelled through internal vibrations. We demonstrate that their dynamics is well captured by the model of active Brownian particles, albeit with an angular speed that differs between individual units. Transferring to a numerical model, we show that this polydispersity of angular speeds gives rise to specific collective behavior: self-sorting under confinement and enhancement of translational diffusion. Our results show that, while naively perceived as imperfection, disorder of individual properties can provide another route to realize programmable active matter.

Non-Brownian diffusion and chaotic rheology of autophoretic disks

The dynamics of a two-dimensional autophoretic disk is quantified as a minimal model for the chaotic trajectories undertaken by active droplets. Via direct numerical simulations, we show that the mean-square displacement of the disk in a quiescent fluid is linear at long times. Surprisingly, however, this apparently diffusive behavior is non-Brownian, owing to strong cross correlations in the displacement tensor. The effect of a shear flow field on the chaotic motion of an autophoretic disk is examined. Here, the stresslet on the disk is chaotic for weak shear flows; a dilute suspension of such disks would exhibit a chaotic shear rheology. This chaotic rheology is quenched first into a periodic state and ultimately a steady state as the flow strength is increased.

Persistent motion of a Brownian particle subject to repulsive feedback with time delay

Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal particle moving in two dimensions under time-delayed, nonlinear feedback control. Specifically, the particle is subject to a force derived from a repulsive Gaussian potential depending on the difference between its instantaneous position, r(t), and its earlier position r(t-τ), where τ is the delay time. Considering first the deterministic case, we provide analytical results for both the case of small displacements and the dynamics at long times. In particular, at appropriate values of the feedback parameters, the particle approaches a steady state with a constant, nonzero velocity whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times, but the (numerically obtained) velocity autocorrelation still reveals some persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a mixed regime at intermediate times with contributions of both ballistic motion and diffusive translational motion, allowing us to extract an estimate for the effective propulsion velocity in presence of noise. We then analyze the data in terms of exact, known results for the MSD of active Brownian particles. The comparison indeed indicates a strong similarity between the dynamics of the particle under repulsive delayed feedback and active motion. This relation carries over to the behavior of the long-time diffusion coefficient D_{eff} which, similarly to active motion, is strongly enhanced compared to the free case. Finally, we show that, for small delays, D_{eff} can be estimated analytically.

Microswimmers in vortices: dynamics and trapping

Biological and artificial microswimmers often self-propel in external flows of vortical nature; relevant examples include algae in small-scale ocean eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic devices. A recent experiment has shown that swimming bacteria in model vortices are expelled from the vortex all the way to a well-defined depletion zone (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114). In this paper, we propose a theoretical model to investigate the dynamics of elongated microswimmers in elementary vortices, namely active particles in two- and three-dimensional rotlets. A deterministic model first reveals the existence of bounded orbits near the centre of the vortex and unbounded orbits elsewhere. We further discover a conserved quantity of motion that allows us to map the phase space according to the type of the orbit (bounded vs unbounded). We next introduce translational and rotational noise into the system. Using a Fokker-Planck formalism, we quantify the quality of trapping near the centre of the vortex by examining the probability of escape and the mean time of escape from the region of deterministically bounded orbits. We finally show how to use these findings to formulate a prediction for the radius of the depletion zone, which compares favourably with the experiments (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114).

Circular motion subject to external alignment under active driving: Nonlinear dynamics and the circle map

Hardly any real self-propelling or actively driven object is perfect. Thus, undisturbed motion will generally not follow straight lines but rather bent or circular trajectories. We here address self-propelled or actively driven objects that move in discrete steps and additionally tend to migrate towards a certain direction by discrete angular adjustment. Overreaction in the angular alignment is possible. This competition implies pronounced nonlinear dynamics including period doubling and chaotic behavior in a broad parameter regime. Such behavior directly affects the appearance of the trajectories. Furthermore, we address collective motion and effects of spatial self-concentration.

Fitting an active Brownian particle's mean-squared displacement with improved parameter estimation

The active Brownian particle (ABP) model is widely used to describe the dynamics of active matter systems, such as Janus microswimmers. In particular, the analytical expression for an ABP's mean-squared displacement (MSD) is useful as it provides a means to describe the essential physics of a self-propelled, spherical Brownian particle. However, the truncated or "short-time" form of the MSD equation is typically fitted, which can lead to significant problems in parameter estimation. Furthermore, heteroscedasticity and the often statistically dependent observations of an ABP's MSD lead to a situation where standard ordinary least-squares regression leads to biased estimates and unreliable confidence intervals. Instead, we propose here to revert to always fitting the full expression of an ABP's MSD at short timescales, using bootstrapping to construct confidence intervals of the fitted parameters. Additionally, after comparison between different fitting strategies, we propose to extract the physical parameters of an ABP using its mean logarithmic squared displacement. These steps improve the estimation of an ABP's physical properties and provide more reliable confidence intervals, which are critical in the context of a growing interest in the interactions of microswimmers with confining boundaries and the influence on their motion.

Nonequilibrium dynamical structure factor of a dilute suspension of active particles in a viscoelastic fluid

In this work we investigate the dynamics of the number-density fluctuations of a dilute suspension of active particles in a linear viscoelastic fluid. We propose a model for the frequency-dependent diffusion coefficient of the active particles which captures the effect of rotational diffusion on the persistence of their self-propelled motion and the viscoelasticity of the medium. Using fluctuating hydrodynamics, the linearized equations for the active suspension are derived, from which we calculate its dynamic structure factor and the corresponding intermediate scattering function. For a Maxwell-type rheological model, we find an intricate dependence of these functions on the parameters that characterize the viscoelasticity of the solvent and the activity of the particles, which can significantly deviate from those of an inert suspension of passive particles and of an active suspension in a Newtonian solvent. In particular, in some regions of the parameter space we uncover the emergence of oscillations in the intermediate scattering function at certain wave numbers which represent the hallmark of the nonequilibrium particle activity in the dynamical structure of the suspension and also encode the viscoelastic properties of the medium.

Liquid-hexatic-solid phases in active and passive Brownian particles determined by stochastic birth and death events

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 (coexisting hexatic and liquid phases) occurs for large activity and sufficiently small birth rates. We also observe a reentrant transition to the hexatic phase when the birth rate is increased. This results from a balance of an increasing number of particles filling the system, and a larger number of defects resulting from the birth and death dynamics.

Inertial dynamics of an active Brownian particle

Active Brownian motion commonly assumes spherical overdamped particles. However, self-propelled particles are often neither symmetric nor overdamped yet underlie random fluctuations from their surroundings. Active Brownian motion has already been generalized to include asymmetric particles. Separately, recent findings have shown the importance of inertial effects for particles of macroscopic size or in low-friction environments. We aim to consolidate the previous findings into the general description of a self-propelled asymmetric particle with inertia. We derive the Langevin equation of such a particle as well as the corresponding Fokker-Planck equation. Furthermore, a formula is presented that allows reconstructing the hydrodynamic resistance matrix of the particle by measuring its trajectory. Numerical solutions of the Langevin equation show that, independently of the particle's shape, the noise-free trajectory at zero temperature starts with an inertial transition phase and converges to a circular helix. We discuss this universal convergence with respect to the helical motion that many microorganisms exhibit.

Diffusion coefficient and power spectrum of active particles with a microscopically reversible mechanism of self-propelling

Catalytically active macromolecules are envisioned as key building blocks in development of artificial nanomotors. However, theory and experiments report conflicting findings regarding their dynamics. The lack of consensus is mostly caused by a limited understanding of specifics of self-propulsion mechanisms at the nanoscale. Here, we study a generic model of a self-propelled nanoparticle that does not rely on a particular mechanism. Instead, its main assumption is the fundamental symmetry of microscopic dynamics of chemical reactions: the principle of microscopic reversibility. Significant consequences of this assumption arise if we subject the particle to an action of an external time-periodic force. The particle diffusion coefficient is then enhanced compared to the unbiased dynamics. The enhancement can be controlled by the force amplitude and frequency. We also derive the power spectrum of particle trajectories. Among new effects stemming from the microscopic reversibility are the enhancement of the spectrum at all frequencies and sigmoid-shaped transitions and a peak at characteristic frequencies of rotational diffusion and external forcing. The microscopic reversibility is a generic property of a broad class of chemical reactions, therefore we expect that the presented results will motivate new experimental studies aimed at testing of our predictions. This could provide new insights into dynamics of catalytic macromolecules.

Statistics for an object actively driven by spontaneous symmetry breaking into reversible directions

Propulsion of otherwise passive objects is achieved by mechanisms of active driving. We concentrate on cases in which the direction of active drive is subject to spontaneous symmetry breaking. In our case, this direction will be maintained until a large enough impulse by an additional stochastic force reverses it. Examples may be provided by self-propelled droplets, gliding bacteria stochastically reversing their propulsion direction, or nonpolar vibrated hoppers. The magnitude of active forcing is regarded as constant, and we include the effect of inertial contributions. Interestingly, this situation can formally be mapped to stochastic motion under (dry, solid) Coulomb friction, however, with a negative friction parameter. Diffusion coefficients are calculated by formal mapping to the situation of a quantum-mechanical harmonic oscillator exposed to an additional repulsive delta-potential. Results comprise a ditched or double-peaked velocity distribution and spatial statistics showing outward propagating maxima when starting from initially concentrated arrangements.

Group formation and collective motion of colloidal rods with an activity triggered by visual perception

We investigate the formation of cohesive groups and the collective diffusion of colloidal spherocylinders with a motility driven by a simple visual perception model. For this, we perform Brownian dynamics simulations without hydrodynamic interactions. The visual perception is based on sight cones attached to the spherocylinders and perception functions quantifying the visual stimuli. If the perception function of a particle reaches a predefined threshold, an active component is added to its motion. We find that, in addition to the opening angle of the cone of sight, the aspect ratio of the particles plays an important role for the formation of cohesive groups. If the elongation of the particles is increased, the maximum angle for which the rods organize themselves into such groups decreases distinctly. After a system forms a cohesive group, it performs a diffusive motion, which can be quantified by an effective diffusion coefficient. For increasing aspect ratios, the spatial expansion of the cohesive groups and the effective diffusion coefficient of the collective motion increase, while the number of active group members decreases. We also find that a larger particle number, a smaller propulsion velocity of the group members, and a smaller threshold for the visual stimulus increase the maximum opening angle for which cohesive groups form. Based on our results, we expect anisotropic particles to be of great relevance for the adjustability of visual perception-dependent motility.

Inertial active Ornstein-Uhlenbeck particle in the presence of a magnetic field

We consider an inertial active Ornstein-Uhlenbeck particle in an athermal bath. The particle is charged, constrained to move in a two-dimensional harmonic trap, and a magnetic field is applied perpendicular to the plane of motion. The steady-state correlations and the mean-square displacement are studied when the particle is confined as well as when it is set free from the trap. With the help of both numerical simulation and analytical calculations, we observe that inertia plays a crucial role in the dynamics in the presence of a magnetic field. In a highly viscous medium where the inertial effects are negligible, the magnetic field has no influence on the correlated behavior of position as well as velocity. In the time asymptotic limit, the overall displacement of the confined harmonic particle gets enhanced by the presence of a magnetic field and saturates for a stronger magnetic field. On the other hand, when the particle is set free, the overall displacement gets suppressed and approaches zero when the strength of the field is very high. Interestingly, it is seen that in the time asymptotic limit, the confined harmonic particle behaves like a passive particle and becomes independent of the activity, especially in the presence of a very strong magnetic field. Similarly, for a free particle the mean-square displacement in the long time limit becomes independent of activity even for a longer persistence of noise cor- relation in the dynamics.

Active Brownian motion with memory delay induced by a viscoelastic medium

By now active Brownian motion is a well-established model to describe the motion of mesoscopic self-propelled particles in a Newtonian fluid. On the basis of the generalized Langevin equation, we present an analytic framework for active Brownian motion with memory delay assuming time-dependent friction kernels for both translational and orientational degrees of freedom to account for the time-delayed response of a viscoelastic medium. Analytical results are obtained for the orientational correlation function, mean displacement, and mean-square displacement which we evaluate in particular for a Maxwell fluid characterized by a kernel which decays exponentially in time. Further, we identify a memory-induced delay between the effective self-propulsion force and the particle orientation which we quantify in terms of a special dynamical correlation function. In principle, our predictions can be verified for an active colloidal particle in various viscoelastic environments such as a polymer solution.

Self-propulsion with speed and orientation fluctuation: Exact computation of moments and dynamical bistabilities in displacement

We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. The speed fluctuation brings about a dynamical anisotropy even in the absence of shape anisotropy. We use the Laplace transform of the Fokker-Planck equation to obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The dynamical anisotropy leads to a subdiffusive scaling in the parallel component of displacement fluctuation at intermediate times. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two dimensions shows two crossovers between compact and extended trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the wormlike chain model of semiflexible polymers.

Mechanically Optimize T Cells Activation by Spiky Nanomotors

T cell activation is vital for immune response initiation and modulation. Except for the strength of the interaction between T cell receptors (TCR) and peptides on major histocompatibility complex molecules (MHC), mechanical force, mediated by professional mechanosensitive ion channels, contributes to activating T cells. The intrinsic characteristic of synthetic micro/nanomotors that convert diverse energy sources into physical movement and force, opening up new possibilities for T cell regulation. In this work, Pd/Au nanomotors with spiky morphology were fabricated, and in the presence of low concentrations of hydrogen peroxide fuel, the motors exhibited continuous locomotion in the cellular biological environment. Physical cues (force and pressure) generated by the dynamic performance are sensed by mechanosensitive ion channels of T cells and trigger Ca2+ influx and subsequent activation. The successful demonstration that mechanical signals generated in the bio microenvironment can potentiate T cells activation, represents a potential approach for cell-based cancer immunotherapy.

Transport and phase separation of active Brownian particles in fluctuating environments

In this work, we study the dynamics of a single active Brownian particle, as well as the collective behavior of interacting active Brownian particles, in a fluctuating heterogeneous environment. We employ a variant of the diffusing diffusivity model where the equation of motion of the active particle involves a time-dependent motility and diffusivities. Within our model, those fluctuations are coupled to each other. Using analytical methods, we obtain the probability distribution function of particle displacement and its moments for a single particle. We then investigate the impact of the environmental fluctuations on the collective behavior of the active Brownian particles by means of extensive numerical simulations. Our results show that the fluctuations hinder the motility-induced phase separation, accompanied by a significant change of the density dependence of particle velocities. These effects are interpreted using our analytical results for the dynamics of a single particle.

Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter

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

Phase separation of self-propelled disks with ferromagnetic and nematic alignment

We present a comprehensive study of a model system of repulsive self-propelled disks in two dimensions with ferromagnetic and nematic velocity alignment interactions. We characterize the phase behavior of the system as a function of the alignment and self-propulsion strength, featuring orientational order for strong alignment and motility-induced phase separation (MIPS) at moderate alignment but high enough self-propulsion. We derive a microscopic theory for these systems yielding a closed set of hydrodynamic equations from which we perform a linear stability analysis of the homogenous disordered state. This analysis predicts MIPS in the presence of aligning torques. The nature of the continuum theory allows for an explicit quantitative comparison with particle-based simulations, which consistently shows that ferromagnetic alignment fosters phase separation, while nematic alignment does not alter either the nature or the location of the instability responsible for it. In the ferromagnetic case, such behavior is due to an increase of the imbalance of the number of particle collisions along different orientations, giving rise to the self-trapping of particles along their self-propulsion direction. On the contrary, the anisotropy of the pair correlation function, which encodes this self-trapping effect, is not significantly affected by nematic torques. Our work shows the predictive power of such microscopic theories to describe complex active matter systems with different interaction symmetries and sheds light on the impact of velocity-alignment interactions in motility-induced phase separation.

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.

Effect of polydispersity on the dynamics of active Brownian particles

We numerically study the dynamics and the phases of self-propelled disk-shaped particles of different sizes with soft repulsive potential in two dimensions. Size diversity is introduced by the polydispersity index (PDI) ε, which is the width of the uniform distribution of the particle's radius. The self-propulsion speed of the particles controls the activity v. We observe enhanced dynamics for large size diversity among the particles. We calculate the effective diffusion coefficient D_{eff} in the steady state. The system exhibits four distinct phases, jammed phase with small D_{eff} for small activity and liquid phase with enhanced D_{eff} for large activity. The number fluctuation is larger and smaller than the equilibrium limit in the liquid and jammed phases, respectively. Further, the jammed phase is of two types: solid jammed and liquid jammed for small and large PDI. Whereas the liquid phase is called motility induced phase separation (MIPS) liquid for small PDI and for large PDI, we find enhanced diffusivity and call it the pure liquid phase. The system is studied for three packing densities ϕ, and the response of the system for polydispersity is the same for all ϕ's. Our study can help understand the behavior of cells of various sizes in a tissue, artificial self-driven granular particles, or living organisms of different sizes in a dense environment.

The control effort to steer self-propelled microswimmers depends on their morphology: comparing symmetric spherical versus asymmetric L-shaped particles

Active goal-directed motion requires real-time adjustment of control signals depending on the system's status, also known as control. The amount of information that needs to be processed depends on the desired motion and control, and on the system's morphology. The morphology of the system may directly effectuate or support the desired motion. This morphology-based reduction to the neuronal 'control effort' can be quantified by a novel information-entropy-based approach. Here, we apply this novel measure of 'control effort' to active microswimmers of different morphology. Their motion is a combination of directed deterministic and stochastic motion. In spherical microswimmers, the active propulsion leads to linear velocities. Active propulsion of asymmetric L-shaped particles leads to circular or-on tilted substrates-directed motion. Thus, the difference in shape, i.e. the morphology of the particles, directly influence the motion. Here, we quantify how this morphology can be exploited by control schemes for the purpose of steering the particles towards targets. Using computer simulations, we found in both cases a significantly lower control effort for L-shaped particles. However, certain movements can only be achieved by spherical particles. This demonstrates that a suitably designed microswimmer's morphology might be exploited to perform specific tasks.

Two-dimensional Brownian motion of anisotropic dimers

We study the two-dimensional motion of colloidal dimers by single-particle tracking and compare the experimental observations obtained by bright-field microscopy to theoretical predictions for anisotropic diffusion. The comparison is based on the mean-square displacements in the laboratory and particle frame as well as generalizations of the self-intermediate scattering functions, which provide insights into the rotational dynamics of the dimer. The diffusional anisotropy leads to a measurable translational-rotational coupling that becomes most prominent by aligning the coordinate system with the initial orientation of the particles. In particular, we find a splitting of the time-dependent diffusion coefficients parallel and perpendicular to the long axis of the dimer which decays over the orientational relaxation time. Deviations of the self-intermediate scattering functions from pure exponential relaxation are small but can be resolved experimentally. The theoretical predictions and experimental results agree quantitatively.

Control and controllability of microswimmers by a shearing flow

With the continuing rapid development of artificial microrobots and active particles, questions of microswimmer guidance and control are becoming ever more relevant and prevalent. In both the applications and theoretical study of such microscale swimmers, control is often mediated by an engineered property of the swimmer, such as in the case of magnetically propelled microrobots. In this work, we will consider a modality of control that is applicable in more generality, effecting guidance via modulation of a background fluid flow. Here, considering a model swimmer in a commonplace flow and simple geometry, we analyse and subsequently establish the efficacy of flow-mediated microswimmer positional control, later touching upon a question of optimal control. Moving beyond idealized notions of controllability and towards considerations of practical utility, we then evaluate the robustness of this control modality to sources of variation that may be present in applications, examining in particular the effects of measurement inaccuracy and rotational noise. This exploration gives rise to a number of cautionary observations, which, overall, demonstrate the need for the careful assessment of both policy and behavioural robustness when designing control schemes for use in practice.

Active noise-driven particles under space-dependent friction in one dimension

We study a Langevin equation describing the stochastic motion of a particle in one dimension with coordinate x, which is simultaneously exposed to a space-dependent friction coefficient γ(x), a confining potential U(x) and nonequilibrium (i.e., active) noise. Specifically, we consider frictions γ(x)=γ_{0}+γ_{1}|x|^{p} and potentials U(x)∝|x|^{n} with exponents p=1,2 and n=0,1,2. We provide analytical and numerical results for the particle dynamics for short times and the stationary probability density functions (PDFs) for long times. The short-time behavior displays diffusive and ballistic regimes while the stationary PDFs display unique characteristic features depending on the exponent values (p,n). The PDFs interpolate between Laplacian, Gaussian, and bimodal distributions, whereby a change between these different behaviors can be achieved by a tuning of the friction strengths ratio γ_{0}/γ_{1}. Our model is relevant for molecular motors moving on a one-dimensional track and can also be realized for confined self-propelled colloidal particles.

Time-dependent inertia of self-propelled particles: The Langevin rocket

Many self-propelled objects are large enough to exhibit inertial effects but still suffer from environmental fluctuations. The corresponding basic equations of motion are governed by active Langevin dynamics, which involve inertia, friction, and stochastic noise for both the translational and orientational degrees of freedom coupled via the self-propulsion along the particle orientation. In this paper, we generalize the active Langevin model to time-dependent parameters and explicitly discuss the effect of time-dependent inertia for achiral and chiral particles. Realizations of this situation are manifold, ranging from minirockets (which are self-propelled by burning their own mass), to dust particles in plasma (which lose mass by evaporating material), to walkers with expiring activity. Here we present analytical solutions for several dynamical correlation functions, such as mean-square displacement and orientational and velocity autocorrelation functions. If the parameters exhibit a slow power law in time, we obtain anomalous superdiffusion with a nontrivial dynamical exponent. Finally, we constitute the "Langevin rocket" model by including orientational fluctuations in the traditional Tsiolkovsky rocket equation. We calculate the mean reach of the Langevin rocket and discuss different mass ejection strategies to maximize it. Our results can be tested in experiments on macroscopic robotic or living particles or in self-propelled mesoscopic objects moving in media of low viscosity, such as complex plasma.

Impact of confined geometries on hopping and trapping of motile bacteria in porous media

We use a random walk particle-tracking (RWPT) approach to elucidate the impact of porous media confinement and cell-cell interactions on bacterial transport. The model employs stochastic alternating motility states consisting of hopping movement and trapping reorientation. The stochastic motility patterns are defined based on direct visualization of individual trajectory data. We validate our model against experimental data, at single-cell resolution, of bacterial E. coli motion in three-dimensional confined porous media. Results show that the model is able to efficiently simulate the spreading dynamics of motile bacteria as it captures the impact of cell-cell interaction and pore confinement, which marks the transition to a late-time subdiffusive regime. Furthermore, the model is able to qualitatively reproduce the observed directional persistence. Our RWPT model constitutes a meshless simple method which is easy to implement and does not invoke ad hoc assumptions but represents the basis for a multiscale approach to the study of bacterial dispersal in porous systems.

Super-Gaussian, superdiffusive transport of multimode active matter

Living matter often exhibits multimode transport that switches between an active, self-propelled motion and a seemingly passive, random motion. Here, we investigate an exactly solvable model of multimode active matter, such as living cells and motor proteins, which alternatingly undergoes active and passive motion. Our model study shows that the reversible transition between a passive mode and an active mode causes super-Gaussian transport dynamics, observed in various experiments. We find the non-Gaussian character of the matter's displacement distribution is essentially determined by the population ratio between active and passive motion. Interestingly, under a certain population ratio of the active and passive modes, the displacement distribution changes from sub-Gaussian to super-Gaussian as time increases. The mean-square displacement of our model exhibits transient superdiffusive dynamics, yet recovers diffusive behavior at both the short- and long-time limits. We finally generalize our model to encompass complex, multimode active matter in an arbitrary spatial dimension.

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

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

Active matter: Quantifying the departure from equilibrium

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

Frictional active Brownian particles

Frictional forces affect the rheology of hard-sphere colloids, at high shear rate. Here we demonstrate, via numerical simulations, that they also affect the dynamics of active Brownian particles and their motility-induced phase separation. Frictional forces increase the angular diffusivity of the particles, in the dilute phase, and prevent colliding particles from resolving their collision by sliding one past to the other. This leads to qualitatively changes of motility-induced phase diagram in the volume-fraction motility plane. While frictionless systems become unstable towards phase separation as the motility increases only if their volume fraction overcomes a threshold, frictional systems become unstable regardless of their volume fraction. These results suggest the possibility of controlling the motility-induced phase diagram by tuning the roughness of the particles.