Memory-Induced Transition from a Persistent Random Walk to Circular Motion for Achiral Microswimmers

Enhanced rotational diffusion and spontaneous rotation of an active Janus disk in a complex fluid

Active colloids and self-propelled particles moving through microstructured fluids can display different behavior compared to what is observed in simple fluids. As they are driven out of equilibrium in complex fluids they can experience enhanced translational and rotational diffusion as well as instabilities. In this work, we study the deterministic and the Brownian rotational dynamics of an active Janus disk propelling at a constant speed through a complex fluid. The interactions between the Janus disk and the complex fluid are modeled using a fluctuating advection-diffusion equation, which we solve using the finite element method. Motivated by experiments, we focus on the case of a complex fluid comprising molecules that are much smaller than the size of the active disk but much bigger than the solvent. Using numerical simulations, we elucidate the interplay between active motion and fluid microstructure that leads to enhanced rotational diffusion and spontaneous rotation observed in experiments employing Janus colloids in polymer solutions. By increasing the propulsion speed of the Janus disk, the simulations predict the onset of a spontaneous rotation and an increase of the rotational diffusion coefficient by orders of magnitude compared to its equilibrium value. These phenomena depend strongly on the number density of the constituents of the complex fluid and their interactions with the two sides of the Janus disk. Given the simplicity of our model, we expect that our findings will apply to a wide range of active systems propelling through complex media.

A computer simulation study of a chiral active ring polymer

We investigate a ring polymer under the influence of chiral active Brownian forces in two dimensions using coarse-grained computer simulations. We observe a non-monotonic behavior of the radius of gyration of an active Brownian ring as a function of active force. However, the shrinkage of the ring in the intermediate strength of active forces becomes more pronounced in the presence of chiral active forces, and the shrinkage is monotonic at a given activity level as a function of the angular frequency controlling the direction of the active force. The distribution of radius of gyration, inter-monomer distance, and radial distribution suggest that the monomers come close to each other, eventually leading to the shrinkage of the ring. Moreover, the bond-correlation suggests that the chirality introduces a local folding of the monomers. Furthermore, using the diameter correlation function, we show that the ring performs tank-treading motion with a frequency following power-law relation with active force with exponent 3/2. The mean squared displacement of the monomers further assists the tank-treading dynamics by exhibiting oscillatory behavior.

Dynamics of non-Markovian systems: Markovian embedding versus effective mass approach

Dynamics of non-Markovian systems is a classic problem yet it attracts everlasting activity in physics and beyond. A powerful tool for modeling such setups is the generalized Langevin equation, however, its analysis typically poses a major challenge even for numerical means. For this reason, various approximations have been proposed over the years that simplify the original model. In this paper, we compare two methods allowing us to tackle this great challenge: (i) the well-known and successful Markovian embedding technique and (ii) the recently developed effective mass approach. We discuss their scope of applicability, numerical accuracy, and computational efficiency. In doing so, we consider a paradigmatic model of a free Brownian particle subjected to power-law correlated thermal noise. We show that when the memory time is short, the effective mass approach offers satisfying precision and typically is much faster than the Markovian embedding. Moreover, the concept of effective mass can be used to find optimal parameters allowing us to reach supreme accuracy and minimal computational cost within the embedding. Our paper therefore provides a blueprint for investigating the dynamics of non-Markovian systems.

Intermediate scattering function of a gravitactic circle swimmer

We analyze gravitaxis of a Brownian circle swimmer by deriving and analytically characterizing the experimentally measurable intermediate scattering function (ISF). To solve the associated Fokker-Planck equation, we use a spectral-theory approach, finding formal expressions in terms of eigenfunctions and eigenvalues of the overdamped-noisy-driven pendulum problem. We further perform a Taylor series of the ISF in the wavevector to extract the cumulants up to the fourth order. We focus on the skewness and kurtosis analyzed for four observation directions in the 2D plane. Validating our findings involves conducting Langevin-dynamics simulations and interpreting the results using a harmonic approximation. The skewness and kurtosis are amplified as the orienting torque approaches the intrinsic angular drift of the circle swimmer from above, highlighting deviations from Gaussian behavior. Transforming the ISF to the comoving frame, a measurable quantity, reveals gravitactic effects and diverse behaviors spanning from diffusive motion at low wavenumbers to circular motion at intermediate wavenumbers and directed motion at higher wavenumbers.

Memory-induced absolute negative mobility

Non-Markovian systems form a broad area of physics that remains greatly unexplored despite years of intensive investigations. The spotlight is on memory as a source of effects that are absent in their Markovian counterparts. In this work, we dive into this problem and analyze a driven Brownian particle moving in a spatially periodic potential and exposed to correlated thermal noise. We show that the absolute negative mobility effect, in which the net movement of the particle is in the direction opposite to the average force acting on it, may be induced by the memory of the setup. To explain the origin of this phenomenon, we resort to the recently developed effective mass approach to dynamics of non-Markovian systems.

Swirling Due to Misaligned Perception-Dependent Motility

A system of particles with motility variable in terms of a vision-type of perception is investigated by a combination of Langevin dynamics simulations in two-dimensional systems and an analytical approach based on conservation law principles. Persistent swirling with predetermined direction is here induced by differentiating the self-propulsion direction and the perception cone axis. Clusters can have a fluidlike center with a rotating outer layer or display a solidlike rotation driven by the outer layer activity. Discontinuous motility with misaligned perception might therefore serve as a powerful self-organization strategy in microrobots.

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.

Memory Corrections to Markovian Langevin Dynamics

Analysis of non-Markovian systems and memory-induced phenomena poses an everlasting challenge in the realm of physics. As a paradigmatic example, we consider a classical Brownian particle of mass M subjected to an external force and exposed to correlated thermal fluctuations. We show that the recently developed approach to this system, in which its non-Markovian dynamics given by the Generalized Langevin Equation is approximated by its memoryless counterpart but with the effective particle mass M∗ Collapse

Key Words

• colored noise
• correlated fluctuations
• memory
• non-Markovian

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MESH Headings Collapse

Grants

• 2022/45/B/ST3/02619 National Science Center

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Affiliation(s)

Mateusz Wiśniewski
Jerzy Łuczka
Jakub Spiechowicz Institute of Physics, University of Silesia, 41-500 Chorzów, Poland

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Emergence of Memory in Equilibrium versus Nonequilibrium Systems

Experiments often probe observables that correspond to low-dimensional projections of high-dimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remain poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom either evolve cooperatively or are coupled by a hidden nonequilibrium current. In the reversible setting the strongest memory manifests when the timescales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear timescale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a "diagnostic" of the underlying hidden transport mechanism.

Effective mass approach to memory in non-Markovian systems

Recent pioneering experiments on non-Markovian dynamics done, e.g., for active matter have demonstrated that our theoretical understanding of this challenging yet hot topic is rather incomplete and there is a wealth of phenomena still awaiting discovery. It is related to the fact that typically for simplification the Markovian approximation is employed and as a consequence the memory is neglected. Therefore, methods allowing to study memory effects are extremely valuable. We demonstrate that a non-Markovian system described by the Generalized Langevin Equation (GLE) for a Brownian particle of mass M can be approximated by the memoryless Langevin equation in which the memory effects are correctly reproduced solely via the effective mass M^{*} of the Brownian particle which is determined only by the form of the memory kernel. Our work lays the foundation for an impactful approach which allows one to readily study memory-related corrections to Markovian dynamics.

Chiral active particles are sensitive reporters to environmental geometry

Chiral active particles (CAPs) are self-propelling particles that break time-reversal symmetry by orbiting or spinning, leading to intriguing behaviors. Here, we examined the dynamics of CAPs moving in 2D lattices of disk obstacles through active Brownian dynamics simulations and granular experiments with grass seeds. We find that the effective diffusivity of the CAPs is sensitive to the structure of the obstacle lattice, a feature absent in achiral active particles. We further studied the transport of CAPs in obstacle arrays under an external field and found a reentrant directional locking effect, which can be used to sort CAPs with different activities. Finally, we demonstrated that parallelogram lattices of obstacles without mirror symmetry can separate clockwise and counter-clockwise CAPs. The mechanisms of the above three novel phenomena are qualitatively explained. As such, our work provides a basis for designing chirality-based tools for single-cell diagnosis and separation, and active particle-based environmental sensors.

Motility and swimming: universal description and generic trajectories

Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that for a particle driven by a concentration field of an active species, straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the particle or that of suspending medium. Our proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations, leading to spontaneous symmetry breaking in the concentration field driving the particle motility. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium, or influence of bounding walls are not necessary.

Direct numerical simulations of a microswimmer in a viscoelastic fluid

This study presents the application of the smoothed profile (SP) method to perform direct numerical simulations for the motion of both passive and active "squirming" particles in Newtonian and viscoelastic fluids. We found that fluid elasticity has a significant impact on both the transient behavior and the steady-state velocity of the particles. Specifically, we observe that the swirling flow generated by the squirmer's surface velocity significantly enhances their swimming speed as the Weissenberg number increases, regardless of the swimming type. Furthermore, we find that pushers outperform pullers in Oldroyd-B fluids, suggesting that the speed of a squirmer depends on the swimmer type. To understand the physical origin of the phenomenon of swirling flow enhancing the swimming speed, we investigate the velocity field and polymer conformation around non-swirling and swirling neutral squirmers in viscoelastic fluids. Our investigation reveals that the velocity field around the neutral swirling squirmers exhibits pusher-like extensional flow characteristics, as well as an asymmetric polymer conformation distribution, which gives rise to this increased propulsion. This is confirmed by the investigation of the force on a fixed squirmer, which revealed that the polymer stress, particularly its diagonal components, plays a critical role in enhancing the swimming speed of swirling squirmers in viscoelastic fluids. Additionally, our results demonstrate that the maximum swimming speeds of swirling squirmers occur at an intermediate value of the fluid viscosity ratio for all swimmer types. These findings have important implications for understanding the behavior of particles and micro-organisms in complex fluids.

Self-Solidifying Active Droplets Showing Memory-Induced Chirality

Most synthetic microswimmers do not reach the autonomy of their biological counterparts in terms of energy supply and diversity of motions. Here, this work reports the first all-aqueous droplet swimmer powered by self-generated polyelectrolyte gradients, which shows memory-induced chirality while self-solidifying. An aqueous solution of surface tension-lowering polyelectrolytes self-solidifies on the surface of acidic water, during which polyelectrolytes are gradually emitted into the surrounding water and induce linear self-propulsion via spontaneous symmetry breaking. The low diffusion coefficient of the polyelectrolytes leads to long-lived chemical trails which cause memory effects that drive a transition from linear to chiral motion without requiring any imposed symmetry breaking. The droplet swimmer is capable of highly efficient removal (up to 85%) of uranium from aqueous solutions within 90 min, benefiting from self-propulsion and flow-induced mixing. These results provide a route to fueling self-propelled agents which can autonomously perform chiral motion and collect toxins.

Dynamics of Active SiO2-Pt Janus Colloids in Dilute Poly(ethylene oxide) Solutions

Self-propelled Janus colloids (JCs) have recently gained much attention due to their ability to move autonomously and mimic biological microswimmers. This ability makes them suitable for prospective drug/cargo-delivery applications in microscopic domains. Understanding their dynamics in surroundings doped with macromolecules such as polymers is crucial, as most of the target application media are complex in nature. In this study, we investigate the self-diffusiophoretic motion of hydrogen peroxide-fuelled SiO₂-Pt JCs in the presence of dilute amounts of poly(ethylene oxide) (PEO). Despite the addition of PEO chains producing a Newtonian behavior with negligible increase in viscosity, the ballistic movement and rotational fluctuations of active JCs are observed to be significantly suppressed. With an increase in the polymer concentration, this leads to a transition from smooth to jittery to cage-hopping to the arrested motion of active JCs. We further propose that the anisotropic interaction of the polymers with the JC increases the "local drag" of the medium, resulting in the unusual impediment of the active motion.

Dynamics and interactions of Quincke roller clusters: From orbits and flips to excited states

Active matter systems may be characterized by the conversion of energy into active motion, e.g., the self-propulsion of microorganisms. Artificial active colloids form models that exhibit essential properties of more complex biological systems but are amenable to laboratory experiments. While most experimental models consist of spheres, active particles of different shapes are less understood. Furthermore, interactions between these anisotropic active colloids are even less explored. Here, we investigate the motion of active colloidal clusters and the interactions between them. We focus on self-assembled dumbbells and trimers powered by an external dc electric field. For dumbbells, we observe an activity-dependent behavior of spinning, circular, and orbital motions. Moreover, collisions between dumbbells lead to the hierarchical self-assembly of tetramers and hexamers, both of which form rotational excited states. On the other hand, trimers exhibit flipping motion that leads to trajectories reminiscent of a honeycomb lattice.

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.

Onsager reciprocal relations and chemo-mechanical coupling for chemically active colloids

Similar to cells, bacteria, and other micro-organisms, synthetic chemically active colloids can harness the energy from their environment through a surface chemical reaction and use the energy to self-propel in fluidic environments. In this paper, we study the chemo-mechanical coupling that leads to the self-propulsion of chemically active colloids. The coupling between chemical reactions and momentum transport is a consequence of Onsager reciprocal relations. They state that the velocity and the surface reaction rate are related to mechanical and chemical affinities through a symmetric matrix. A consequence of Onsager reciprocal relations is that if a chemical reaction drives the motion of the colloid, then an external force generates a reaction rate. Here, we investigate Onsager reciprocal relations for a spherical active colloid that catalyzes a reversible surface chemical reaction between two species. We solve the relevant transport equations using a perturbation expansion and numerical simulations to demonstrate the validity of reciprocal relations around the equilibrium. Our results are consistent with previous studies and highlight the key role of solute advection in preserving the symmetry of the Onsager matrix. Finally, we show that Onsager reciprocal relations break down around a nonequilibrium steady state, which has implications for the thermal fluctuations of the active colloids used in experiments.

Understanding enhanced rotational dynamics of active probes in rod suspensions

Active Brownian particles (APs) have recently been shown to exhibit enhanced rotational diffusion (ERD) in complex fluids. Here, we experimentally observe ERD and numerically corroborate its microscopic origin for a quasi-two-dimensional suspension of colloidal rods. At high density, the rods form small rafts, wherein they perform small-amplitude, high-frequency longitudinal displacements. Activity couples AP-rod contacts to reorientation, with the variance therein leading to ERD. This is captured by a local, rather than a global relaxation time, as used in previous phenomenological modeling. Our result should prove relevant to the microrheological characterization of complex fluids and furthering our understanding of the dynamics of microorganisms in such media.

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.

Memory effects in spiral diffusion of rotary self-propellers

The coupling of deterministic rotary motion and stochastic orientational diffusion of a self-propeller leads to a spiral trajectory of the expected displacement. We extend our former analysis of spiral diffusion [Phys. Rev. E 94, 030601(R) (2016)10.1103/PhysRevE.94.030601] in the white-noise limit to a more realistic scenario of stochastic noise with Gaussian memory and orientational fluctuations driven by an Ornstein-Uhlenbeck process. A variety of dynamical regimes including crossovers from ballistic to diffusive to ballistic in the angular dynamics are determined by the inertial timescale, orientational diffusivity, and angular speed.

Mode-coupling theory for tagged-particle motion of active Brownian particles

We derive a mode-coupling theory (MCT) to describe the dynamics of a tracer particle that is embedded in a dense system of active Brownian particles (ABPs) in two spatial dimensions. The ABP undergo translational and rotational Brownian motion and are equipped with a fixed self-propulsion speed along their orientational vector that describes their active motility. The resulting equations of motion for the tagged-particle density-correlation functions describe the various cases of tracer dynamics close to the glass transition: that of a single active particle in a glass-forming passive host suspensions, that of a passive colloidal particle in a suspension of ABP, and that of active tracers in a bath of active particles. Numerical results are presented for these cases assuming hard-sphere interactions among the particles. The qualitative and quantitative accuracy of the theory is tested against event-driven Brownian dynamics (ED-BD) simulations of active and passive hard disks. Simulation and theory are found in quantitative agreement, provided one adjusts the overall density (as known from the passive description of glassy dynamics), and allows for a rescaling of self-propulsion velocities in the active host system. These adjustments account for the fact that ABP-MCT generally overestimates the tendency for kinetic arrest. We confirm in the simulations a peculiar feature of the transient and stationary dynamical density-correlation functions regarding their lack of symmetry under time reversal, demonstrating the nonequilibrium nature of the system and how it manifests itself in the theory.

Spontaneous chiralization of polar active particles

Polar active particles constitute a wide class of active matter that is able to propel along a preferential direction, given by their polar axis. Here, we demonstrate a generic active mechanism that leads to their spontaneous chiralization through a symmetry-breaking instability. We find that the transition of an active particle from a polar to a chiral symmetry is characterized by the emergence of active rotation and of circular trajectories. The instability is driven by the advection of a solute that interacts differently with the two portions of the particle surface and it occurs through a supercritical pitchfork bifurcation.

Orbital magnetism of an active particle in viscoelastic suspension

We consider an active (self-propelling) particle in a viscoelastic fluid. The particle is charged and constrained to move in a two-dimensional harmonic trap. Its dynamics is coupled to a constant magnetic field applied perpendicular to its plane of motion via Lorentz force. Due to the finite activity, the generalized fluctuation-dissipation relation (GFDR) breaks down, driving the system away from equilibrium. While breaking GFDR, we have shown that the system can have finite classical orbital magnetism only when the dynamics of the system contains finite inertia. The orbital magnetic moment has been calculated exactly. Remarkably, we find that when the elastic dissipation timescale of the medium is larger (smaller) than the persistence timescale of the self-propelling particle, it is diamagnetic (paramagnetic). Therefore, for a given strength of the magnetic field, the system undergoes a transition from diamagnetic to paramagnetic state (and vice versa) simply by tuning the timescales of underlying physical processes, such as active fluctuations and viscoelastic dissipation. Interestingly, we also find that the magnetic moment, which vanishes at equilibrium, behaves nonmonotonically with respect to increasing persistence of self-propulsion, which drives the system out of equilibrium.

Work fluctuation relation of an active Brownian particle in a viscoelastic fluid

We experimentally investigate the work fluctuations of an active Brownian particle (ABP) during its self-propelled motion in a viscoelastic medium. Under such conditions, ABPs display a persistent circular motion which allows the determination of the orientational work fluctuations along its trajectory. Due to the nonlinear coupling to the non-Markovian bath, we find strong deviations from the work fluctuation theorem (WFT) due to observed increased rotational ABP dynamics. Taking this enhanced rotational diffusion into account, the orientational work distributions can be recasted to be in accordance with the WFT by considering an effective temperature of about two orders of magnitude larger than k_{B}T. This approach is confirmed by the good agreement of the torque exerted by the viscoelastic bath on the ABP obtained from the WFT with the value obtained from the mean angular velocity and the friction coefficient of the ABP.

Edge current and pairing order transition in chiral bacterial vortices

Bacterial suspensions show turbulence-like spatiotemporal dynamics and vortices moving irregularly inside the suspensions. Understanding these ordered vortices is an ongoing challenge in active matter physics, and their application to the control of autonomous material transport will provide significant development in microfluidics. Despite the extensive studies, one of the key aspects of bacterial propulsion has remained elusive: The motion of bacteria is chiral, i.e., it breaks mirror symmetry. Therefore, the mechanism of control of macroscopic active turbulence by microscopic chirality is still poorly understood. Here, we report the selective stabilization of chiral rotational direction of bacterial vortices in achiral circular microwells sealed by an oil/water interface. The intrinsic chirality of bacterial swimming near the top and bottom interfaces generates chiral collective motions of bacteria at the lateral boundary of the microwell that are opposite in directions. These edge currents grow stronger as bacterial density increases, and, within different top and bottom interfaces, their competition leads to a global rotation of the bacterial suspension in a favored direction, breaking the mirror symmetry of the system. We further demonstrate that chiral edge current favors corotational configurations of interacting vortices, enhancing their ordering. The intrinsic chirality of bacteria is a key feature of the pairing order transition from active turbulence, and the geometric rule of pairing order transition may shed light on the strategy for designing chiral active matter.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Circular swimming motility and disordered hyperuniform state in an algae system

Active matter comprises individually driven units that convert locally stored energy into mechanical motion. Interactions between driven units lead to a variety of nonequilibrium collective phenomena in active matter. One of such phenomena is anomalously large density fluctuations, which have been observed in both experiments and theories. Here we show that, on the contrary, density fluctuations in active matter can also be greatly suppressed. Our experiments are carried out with marine algae ([Formula: see text]), which swim in circles at the air-liquid interfaces with two different eukaryotic flagella. Cell swimming generates fluid flow that leads to effective repulsions between cells in the far field. The long-range nature of such repulsive interactions suppresses density fluctuations and generates disordered hyperuniform states under a wide range of density conditions. Emergence of hyperuniformity and associated scaling exponent are quantitatively reproduced in a numerical model whose main ingredients are effective hydrodynamic interactions and uncorrelated random cell motion. Our results demonstrate the existence of disordered hyperuniform states in active matter and suggest the possibility of using hydrodynamic flow for self-assembly in active matter.

Characterization of MIPS in a suspension of repulsive active Brownian particles through dynamical features

We study a two-dimensional system composed by Active Brownian Particles (ABPs), focusing on the onset of Motility Induced Phase Separation (MIPS), by means of molecular dynamics simulations. For a pure hard-disk system with no translational diffusion, the phase diagram would be completely determined by their density and Péclet number. In our model, two additional effects are present: translational noise and the overlap of particles; we study the effects of both in the phase space. As we show, the second effect can be mitigated if we use, instead of the standard Weeks-Chandler-Andersen potential, a stiffer potential: the pseudo-hard sphere potential. Moreover, in determining the boundary of our phase space, we explore different approaches to detect MIPS and conclude that observing dynamical features, via the non-Gaussian parameter, is more efficient than observing structural ones, such as through the local density distribution function. We also demonstrate that the Vogel-Fulcher equation successfully reproduces the decay of the diffusion as a function of density, with the exception of very high densities. Thus, in this regard, the ABP system behaves similar to a fragile glass.

Active colloids orbiting giant vesicles

Living or artificial self-propelled colloidal particles show original dynamics when they interact with other objects like passive particles, interfaces or membranes. These active colloids can transport small cargos or can be guided by passive objects, performing simple tasks that could be implemented in more complex systems. Here, we present an experimental investigation at the single particle level of the interaction between isolated active colloids and giant unilamellar lipid vesicles. We observed a persistent orbital motion of the active particle around the vesicle, which is independent of both the particle and the vesicle sizes. Force and torque transfers between the active particle and the vesicle is also described. These results differ in many aspects from recent theoretical and experimental reports on active particles interacting with solid spheres or liquid drops, and may be relevant for the study of swimming particles interacting with cells in biology or with microplastics in environmental science.

Activity pulses induce spontaneous flow reversals in viscoelastic environments

Complex interactions between cellular systems and their surrounding extracellular matrices are emerging as important mechanical regulators of cell functions, such as proliferation, motility and cell death, and such cellular systems are often characterized by pulsating actomyosin activities. Here, using an active gel model, we numerically explore spontaneous flow generation by activity pulses in the presence of a viscoelastic medium. The results show that cross-talk between the activity-induced deformations of the viscoelastic surroundings and the time-dependent response of the active medium to these deformations can lead to the reversal of spontaneously generated active flows. We explain the mechanism behind this phenomenon based on the interaction between the active flow and the viscoelastic medium. We show the importance of relaxation time scales of both the polymers and the active particles and provide a phase space over which such spontaneous flow reversals can be observed. Our results suggest new experiments investigating the role of controlled pulses of activity in living systems ensnared in complex mircoenvironments.

Fluid Viscoelasticity Triggers Fast Transitions of a Brownian Particle in a Double Well Optical Potential

Thermally activated transitions are ubiquitous in nature, occurring in complex environments which are typically conceived as ideal viscous fluids. We report the first direct observations of a Brownian bead transiting between the wells of a bistable optical potential in a viscoelastic fluid with a single long relaxation time. We precisely characterize both the potential and the fluid, thus enabling a neat comparison between our experimental results and a theoretical model based on the generalized Langevin equation. Our findings reveal a drastic amplification of the transition rates compared to those in a Newtonian fluid, stemming from the relaxation of the fluid during the particle crossing events.

Active colloids under geometrical constraints in viscoelastic media

We study the behavior of active particles (APs) moving in a viscoelastic fluid in the presence of geometrical confinements. Upon approaching a flat wall, we find that APs slow down due to compression of the enclosed viscoelastic fluid. In addition, they receive a viscoelastic torque leading to sudden orientational changes and departure from walls. Based on these observations, we develop a numerical model which can also be applied to other geometries and yields good agreement with experimental data. Our results demonstrate, that APs are able to move through complex geometrical structures more effectively when suspended in a viscoelastic compared to a Newtonian fluid.

Bayesian inference of the viscoelastic properties of a Jeffrey's fluid using optical tweezers

Bayesian inference is a conscientious statistical method which is successfully used in many branches of physics and engineering. Compared to conventional approaches, it makes highly efficient use of information hidden in a measured quantity by predicting the distribution of future data points based on posterior information. Here we apply this method to determine the stress-relaxation time and the solvent and polymer contributions to the frequency dependent viscosity of a viscoelastic Jeffrey's fluid by the analysis of the measured trajectory of an optically trapped Brownian particle. When comparing the results to those obtained from the auto-correlation function, mean-squared displacement or the power spectrum, we find Bayesian inference to be much more accurate and less affected by systematic errors.

Autonomously Probing Viscoelasticity in Disordered Suspensions

Recent experiments show a strong rotational diffusion enhancement for self-propelled microrheological probes in colloidal glasses. Here, we provide microscopic understanding using simulations with a frictional probe-medium coupling that converts active translation into rotation. Diffusive enhancement emerges from the medium's disordered structure and peaks at a second-order transition in the number of contacts. Our results reproduce the salient features of the colloidal glass experiment and support an effective description that is applicable to a broader class of viscoelastic suspensions.

Translational and rotational dynamics of a self-propelled Janus probe in crowded environments

We computationally investigate the dynamics of a self-propelled Janus probe in crowded environments. The crowding is caused by the presence of viscoelastic polymers or non-viscoelastic disconnected monomers. Our simulations show that the translational as well as rotational mean square displacements have a distinctive three-step growth for fixed values of self-propulsion force, and steadily increase with self-propulsion, irrespective of the nature of the crowder. On the other hand, in the absence of crowders, the rotational dynamics of the Janus probe is independent of self-propulsion force. On replacing the repulsive polymers with sticky ones, translational and rotational mean square displacements of the Janus probe show a sharp drop. Since different faces of a Janus particle interact differently with the environment, we show that the direction of self-propulsion also affects its dynamics. The ratio of long-time translational and rotational diffusivities of the self-propelled probe with a fixed self-propulsion, when plotted against the area fraction of the crowders, passes through a minimum and at higher area fraction merges to its value in the absence of the crowder. This points towards the decoupling of the translational and rotational dynamics of the self-propelled probe at an intermediate area fraction of the crowders. However, such translational-rotational decoupling is absent for passive probes.

Transient coarsening and the motility of optically heated Janus colloids in a binary liquid mixture

A gold-capped Janus particle suspended in a near-critical binary liquid mixture can self-propel under illumination. We have immobilized such a particle in a narrow channel and carried out a combined experimental and theoretical study of the non-equilibrium dynamics of a binary solvent around it - lasting from the very moment of switching illumination on until the steady state is reached. In the theoretical study we use both a purely diffusive and a hydrodynamic model, which we solve numerically. Our results demonstrate a remarkable complexity of the time evolution of the concentration field around the colloid. This evolution is governed by the combined effects of the temperature gradient and the wettability, and crucially depends on whether the colloid is free to move or is trapped. For the trapped colloid, all approaches indicate that the early time dynamics is purely diffusive and characterized by composition layers travelling with constant speed from the surface of the colloid into the bulk of the solvent. Subsequently, hydrodynamic effects set in. Anomalously large nonequilibrium fluctuations, which result from the temperature gradient and the vicinity of the critical point of the binary liquid mixture, give rise to strong concentration fluctuations in the solvent and to permanently changing coarsening patterns not observed for a mobile particle. The early time dynamics around initially still Janus colloids produces a force which is able to set the Janus colloid into motion. The propulsion due to this transient dynamics is in the direction opposite to that observed after the steady state is attained.

Traveling bands, clouds, and vortices of chiral active matter

We consider stochastic dynamics of self-propelled particles with nonlocal normalized alignment interactions subject to phase lag. The role of the lag is to indirectly generate chirality into particle motion. To understand large-scale behavior, we derive a continuum description of an active Brownian particle flow with macroscopic scaling in the form of a partial differential equation for a one-particle probability density function. Due to indirect chirality, we find a spatially homogeneous nonstationary analytic solution for this class of equations. Our development of kinetic and hydrodynamic theories towards such a solution reveals the existence of a wide variety of spatially nonhomogeneous patterns reminiscent of traveling bands, clouds, and vortical structures of linear active matter. Our model may thereby serve as the basis for understanding the nature of chiral active media and designing multiagent swarms with designated behavior.

The 2020 motile active matter roadmap

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of 'active matter' in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano- and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter comprises a major challenge. Hence, to advance, and eventually reach a comprehensive understanding, this important research area requires a concerted, synergetic approach of the various disciplines. The 2020 motile active matter roadmap of Journal of Physics: Condensed Matter addresses the current state of the art of the field and provides guidance for both students as well as established scientists in their efforts to advance this fascinating area.

Enhanced Rotational Motion of Spherical Squirmer in Polymer Solutions

The rotational diffusive motion of a self-propelled, attractive spherical colloid immersed in a solution of self-avoiding polymers is studied by mesoscale hydrodynamic simulations. A drastic enhancement of the rotational diffusion by more than an order of magnitude in the presence of activity is obtained. The amplification is a consequence of two effects, a decrease of the amount of adsorbed polymers by active motion and an asymmetric encounter with polymers on the squirmer surface, which yields an additional torque and random noise for the rotational motion. Our simulations suggest a way to control the rotational dynamics of squirmer-type microswimmers by the degree of polymer adsorption and system heterogeneity.

Two-dimensional active motion

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

Motor and Rotor in One: Light-Active ZnO/Au Twinned Rods of Tunable Motion Modes

Precise control of the motion of micromachines is the key to achieving their functions for practical applications. The main challenge is that a given micromachine can typically exhibit only one motion mode, i.e., translation or rotation, while having multiple modes of motion resulting from a simple actuation is still rare. Here we designed and synthesized photochemically powered zinc oxide/gold (ZnO/Au) rods that exhibit multiple motion modes. Under homogeneous UV irradiation, these ZnO/Au rods undergo a transition from ballistic motion to persistent rotational motion upon increasing the fuel concentration or the light intensity. In addition, the rods can switch modes from a circular motion to a helical motion and then a straight-line motion by tuning the angle of incident light. We envision that such attractive colloidal micromachines with controllable motions hold considerable promise for diverse practical applications.

Diffusiophoresis of active colloids in viscoelastic media

Self-diffusiophoresis of synthetic Janus (Si/Pt) microspheres in the presence of hydrogen peroxide in complex environments is here investigated. We aim to address the single particle dynamics of these active colloids in different viscoelastic fluids. Experimentally, the Janus colloids were dispersed in a dilute polyvinylpyrrolidone (PVP) solution and in a polyacrylamide (PAM) solution in semi-dilute and semi-dilute entangled regime to analyze their Brownian and active motion. These two systems were chosen to probe different relaxation times from relatively short (∼5 ms) for PVP to large (∼14.5 s) for PAM but always smaller than the rotary Brownian motion time scale. Within this regime, we investigate the coupling between the self-propulsion velocity and the medium rheology. Janus particles are found to get physically confined by polymeric entanglements but surprisingly they are able to escape the physical cage in a time scale much shorter than the relaxation time of the polymer solution. This is particularly relevant for application of self-propelling particles in biomedicine, water and soil remediation where complex environments are naturally present.

Chaotic Swimming of Phoretic Particles

The swimming of a rigid phoretic particle in an isotropic fluid is studied numerically as a function of the dimensionless solute emission rate (or Péclet number Pe). The particle sets into motion at a critical Pe. Whereas the particle trajectory is straight at a small enough Pe, it is found that it loses its stability at a critical Pe in favor of a meandering motion. When Pe is increased further, the particle meanders at a short scale but its trajectory wraps into a circle at a larger scale. Increasing even further, Pe causes the swimmer to escape momentarily the circular trajectory in favor of chaotic motion, which lasts for a certain time, before regaining a circular trajectory, and so on. The chaotic bursts become more and more frequent as Pe increases, until the trajectory becomes fully chaotic, via the intermittency scenario. The statistics of the trajectory is found to be of the run-and-tumble-like nature at a short enough time and of diffusive nature at a long time without any source of noise.

Tuning the Random Walk of Active Colloids: From Individual Run-and-Tumble to Dynamic Clustering

Active particles such as swimming bacteria or self-propelled colloids spontaneously self-organize into large-scale dynamic structures. The emergence of these collective states from the motility pattern of the individual particles, typically a random walk, is yet to be probed in a well-defined synthetic system. Here, we report the experimental realization of tunable colloidal motion that reproduces run-and-tumble and Lévy trajectories. We utilize the Quincke effect to achieve controlled sequences of repeated particle runs and random reorientations. We find that a population of these random walkers exhibit behaviors reminiscent of bacterial suspensions such as dynamic clusters and mesoscale turbulentlike flows.

Generalized Ornstein-Uhlenbeck model for active motion

We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E 90, 012111 (2014)10.1103/PhysRevE.90.012111]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.

Illumination-induced motion of a Janus nanoparticle in binary solvents

Using a fluid particle dynamics method we numerically investigate the motion of a spherical Janus particle suspended in a binary liquid mixture, which emerges under heating of one-half of a colloid surface. The method treats simultaneously the flow of the solvent and the motion of the particle, hence, the velocity of the particle can be computed directly. Our approach accounts for a phenomenon of critical adsorption, therefore, a particle that is adsorptionwise nonneutral is always completely covered by an adsorption layer (droplet). In order to establish the mechanism of self-propulsion, we study systematically various combinations of adsorption preference on both hemispheres of the Janus colloid as function of the heating power for symmetric and nonsymmetric binary solvents and for various particle sizes in three spatial dimensions. Only for a particle for which the heated hemisphere is neutral whereas the other hemisphere prefers one of the two components of the mixture does the reversal of the direction of motion occur. The particle self-propels much faster in nonsymmetric binary solvents. Self-propulsion originates from a gradient of mechanical stress, in a way similar to the Marangoni effect. This stress is not localized at the edge but distributed within the whole droplet. We compare our findings with the experimental observations and other theoretical results.