Pressure and flow of exponentially self-correlated active particles

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

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

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

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

Time-reversal symmetries and equilibriumlike Langevin equations

Graham has shown [Z. Phys. B 26, 397 (1977)0340-224X10.1007/BF01570750] that a fluctuation-dissipation relation can be imposed on a class of nonequilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may lose its time-reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time-reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.

Harvesting information to control nonequilibrium states of active matter

We propose to use a correlated noise bath to drive an optically trapped Brownian particle that mimics active biological matter. Due to the flexibility and precision of our setup, we are able to control the different parameters that drive the stochastic motion of the particle with unprecedented accuracy, thus reaching strongly correlated regimes that are not easily accessible with real active matter. In particular, by using the correlation time (i.e., the "color") of the noise as a control parameter, we can trigger transitions between two nonequilibrium steady states with no expended work, but only a calorific cost. Remarkably, the measured heat production is directly proportional to the spectral entropy of the correlated noise, in a fashion that is reminiscent of Landauer's principle. Our procedure can be viewed as a method for harvesting information from the active fluctuations.

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 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.

Tethered tracer in a mixture of hot and cold Brownian particles: can activity pacify fluctuations?

We study how an interacting mixture of components with differing levels of activity can affect the fluctuations of an embedded object such as a tracer. In particular, we consider a simple model of a tracer that is harmonically bound within a mixture of hot and cold Brownian particles, which, like a mixture of active and passive particles, can phase separate. By measuring the fluctuations of the tracer, we find that this collective behavior gives rise to an effective temperature for the tracer. Additionally, we find that there is an increased tendency for cold particles to accumulate on the surface of the tracer due to the hot particles, potentially dampening its fluctuations and decreasing its effective temperature. These results suggest that the phase separation of a mixture of hot/cold or active/passive particles may have strong effects on the fluctuations of an embedded object. We discuss potential implications of these results for experiments on fluctuations of nuclear envelope affected by the activity in the chromatin.

Effect of boundaries on noninteracting weakly active particles in different geometries

We study analytically how noninteracting weakly active particles, for which passive Brownian diffusion cannot be neglected and activity can be treated perturbatively, distribute and behave near boundaries in various geometries. In particular, we develop a perturbative approach for the model of active particles driven by an exponentially correlated random force (active Ornstein-Uhlenbeck particles). This approach involves a relatively simple expansion of the distribution in powers of the Péclet number and in terms of Hermite polynomials. We use this approach to cleanly formulate boundary conditions, which allows us to study weakly active particles in several geometries: confinement by a single wall or between two walls in 1D, confinement in a circular or wedge-shaped region in 2D, motion near a corrugated boundary, and, finally, absorption onto a sphere. We consider how quantities such as the density, pressure, and flow of the active particles change as we gradually increase the activity away from a purely passive system. These results for the limit of weak activity help us gain insight into how active particles behave in the presence of various types of boundaries.

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

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

Surface Tensions between Active Fluids and Solid Interfaces: Bare vs Dressed

We analyze the surface tension exerted at the interface between an active fluid and a solid boundary in terms of tangential forces. Focusing on active systems known to possess an equation of state for the pressure, we show that interfacial forces are of a more complex nature. Using a number of macroscopic setups, we show that the surface tension is a combination of an equation-of-state abiding part and of setup-dependent contributions. The latter arise from generic setup-dependent steady currents which "dress" the measurement of the "bare" surface tension. The former shares interesting properties with its equilibrium counterpart, and can be used to generalize the Young-Laplace law to active systems. Finally, we show how a suitably designed probe can directly access this bare surface tension, which can also be computed using a generalized virial formula.

Active noise experienced by a passive particle trapped in an active bath

We study the properties of active noise experienced by a passive particle harmonically trapped in an active bath. The active bath is either explicitly simulated by an ensemble of active Brownian particles or abstractly represented by an active colored noise in theory. Assuming the equivalence of the two descriptions of the active bath, the active noise in the simulation system, which is directly extracted by fitting theoretical predictions to simulation measurements, is shown to depend on the constraint suffered by the passive tracer. This scenario is in significant contrast to the case of thermal noise that is independent of external trap potentials. The constraint dependence of active noise arises from the fact that the persistent force on the passive particle from the active bath can be influenced by the particle relaxation dynamics. Moreover, due to the interplay between the active collisions and particle relaxation dynamics, the effective temperature of the passive tracer quantified as the ratio of fluctuation to dissipation increases as the constraint strengthens, while the average potential and kinetic energies of the passive particle both decrease.

Stochastic thermodynamics for self-propelled particles

We propose a generalization of stochastic thermodynamics to systems of active particles, which move under the combined influence of stochastic internal self-propulsions (activity) and a heat bath. The main idea is to consider joint trajectories of particles' positions and self-propulsions. It is then possible to exploit formal similarity of an active system and a system consisting of two subsystems interacting with different heat reservoirs and coupled by a nonsymmetric interaction. The resulting thermodynamic description closely follows the standard stochastic thermodynamics. In particular, total entropy production, Δs_{tot}, can be decomposed into housekeeping, Δs_{hk}, and excess, Δs_{ex}, parts. Both Δs_{tot} and Δs_{hk} satisfy fluctuation theorems. The average rate of the steady-state housekeeping entropy production can be related to the violation of the fluctuation-dissipation theorem via a Harada-Sasa relation. The excess entropy production enters into a Hatano-Sasa-like relation, which leads to a generalized Clausius inequality involving the change of the system's entropy and the excess entropy production. Interestingly, although the evolution of particles' self-propulsions is free and uncoupled from that of their positions, nontrivial steady-state correlations between these variables lead to the nonzero excess dissipation in the reservoir coupled to the self-propulsions.

Active particles under confinement and effective force generation among surfaces

We consider the effect of geometric confinement on the steady-state properties of a one-dimensional active suspension subject to thermal noise. The random active force is modeled by an Ornstein-Uhlenbeck process and the system is studied both numerically, by integrating the Langevin governing equations, and analytically by solving the associated Fokker-Planck equation under suitable approximations. The comparison between the two approaches displays a fairly good agreement and in particular, we show that the Fokker-Planck approach can predict the structure of the system both in the wall region and in the bulk-like region where the surface forces are negligible. The simultaneous presence of thermal noise and active forces determines the formation of a layer, extending from the walls towards the bulk, where the system exhibits polar order. We relate the presence of such ordering to the mechanical pressure exerted on the container's walls and show how it depends on the separation of the boundaries and determines a Casimir-like attractive force mediated by the active suspension.

Autonomous Actuation of Zero Modes in Mechanical Networks Far from Equilibrium

A zero mode, or floppy mode, is a nontrivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this Letter, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by nonequilibrium activity. We find that a correlated active bath actuates an infinitesimal zero mode while simultaneously suppressing fluctuations in higher modes compared to thermal fluctuations, which we experimentally mimic by high frequency shaking of a physical network. Furthermore, self-propulsive dynamics spontaneously mobilize finite mechanisms as exemplified by a self-propelled topological soliton. Nonequilibrium activity thus enables autonomous actuation of coordinated mechanisms engineered through network topology.

Generic Long-Range Interactions Between Passive Bodies in an Active Fluid

A single nonspherical body placed in an active fluid generates currents via breaking of time-reversal symmetry. We show that, when two or more passive bodies are placed in an active fluid, these currents lead to long-range interactions. Using a multipole expansion, we characterize their leading-order behaviors in terms of single-body properties and show that they decay as a power law with the distance between the bodies, are anisotropic, and do not obey an action-reaction principle. The interactions lead to rich dynamics of the bodies, illustrated by the spontaneous synchronized rotation of pinned nonchiral bodies and the formation of traveling bound pairs. The occurrence of these phenomena depends on tunable properties of the bodies, thus opening new possibilities for self-assembly mediated by active fluids.

Memory effects in active particles with exponentially correlated propulsion

The Ornstein-Uhlenbeck particle (OUP) model imagines a microscopic swimmer propelled by an active force which is correlated with itself on a finite time scale. Here we investigate the influence of external potentials on an ideal suspension of OUPs, in both one and two spatial dimensions, with particular attention paid to the pressure exerted on "confining walls." We employ a mathematical connection between the local density of OUPs and the statistics of their propulsion force to demonstrate the existence of an equation of state in one dimension. In higher dimensions we show that active particles generate a nonconservative force field in the surrounding medium. A simplified far-from-equilibrium model is proposed to account for OUP behavior in the vicinity of potentials. Building on this, we interpret simulations of OUPs in more complicated situations involving asymmetrical and spatially curved potentials, and characterize the resulting inhomogeneous stresses in terms of competing active length scales.

Statistical mechanics of transport processes in active fluids: Equations of hydrodynamics

The equations of hydrodynamics including mass, linear momentum, angular momentum, and energy are derived by coarse-graining the microscopic equations of motion for systems consisting of rotary dumbbells driven by internal torques. In deriving the balance of linear momentum, we find that the symmetry of the stress tensor is broken due to the presence of non-zero torques on individual particles. The broken symmetry of the stress tensor induces internal spin in the fluid and leads us to consider the balance of internal angular momentum in addition to the usual moment of momentum. In the absence of spin, the moment of momentum is the same as the total angular momentum. In deriving the form of the balance of total angular momentum, we find the microscopic expressions for the couple stress tensor that drives the spin field. We show that the couple stress contains contributions from both intermolecular interactions and the active forces. The presence of spin leads to the idea of balance of moment of inertia due to the constant exchange of particles in a small neighborhood around a macroscopic point. We derive the associated balance of moment of inertia at the macroscale and identify the moment of inertia flux that induces its transport. Finally, we obtain the balances of total and internal energy of the active fluid and identify the sources of heat and heat fluxes in the system.