Generalized Ornstein-Uhlenbeck model for active motion

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.

The effects of asymmetry in active noises on the efficiency of single colloidal Stirling engines with active noises

Molecular machines, which operate in highly fluctuating environments far from equilibrium, may benefit from their non-equilibrium environments. It is, however, a topic of controversy how the efficiency of the microscopic engines can be enhanced. Recent experiments showed that microscopic Stirling engines in bacterial reservoirs could show high performance beyond the equilibrium thermodynamics. In this work, we perform overdamped Langevin dynamics simulations for microscopic Stirling heat engines in bacterial reservoirs and show that the temperature dependence of the magnitude of active noises should be responsible for such high efficiency. Only when we introduce temperature-dependent active noises, the efficiency of the microscopic Stirling engines is enhanced significantly as in experiments.

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.

Entropy production of active particles formulated for underdamped dynamics

The present work investigates the effect of inertia on the entropy production rate Π for all canonical models of active particles for different dimensions and the type of confinement. To calculate Π, the link between the entropy production and dissipation of heat rate is explored, resulting in a simple and intuitive expression. By analyzing the Kramers equation, alternative formulations of Π are obtained and the virial theorem for active particles is derived. Exact results are obtained for particles in an unconfined environment and in a harmonic trap. In both cases, Π is independent of temperature. For the case of a harmonic trap, Π attains a maximal value for τ=ω^{-1}, where τ is the persistence time and ω is the natural frequency of an oscillator. For active particles in one-dimensional box, or other nonharmonic potentials, thermal fluctuations are found to reduce Π.

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.

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.

Motion of a self-propelled particle with rotational inertia

Overdamped active Brownian motion of self-propelled particles in a liquid has been fairly well studied. However, there are a variety of situations in which the overdamped approximation is not justified, for instance, when self-propelled particles move in a low-viscosity medium or when their rotational diffusivity is enhanced by internal active processes or external control. Examples of various origins include biofilaments driven by molecular motors, living and artificial microflyers and interfacial surfers, field-controlled and superfluid microswimmers, vibration-driven granular particles and autonomous mini-robots with sensorial delays, etc. All of them extend active Brownian motion to the underdamped case, i.e., to active Langevin motion, which takes into account inertia. Despite a rich experimental background, there is a gap in the theory in the field where rotational inertia significantly affects the random walk of active particles on all time scales. In particular, although the well-known models of active Brownian and Ornstein-Uhlenbeck particles include a memory effect of the direction of motion, they are not applicable in the underdamped case, because the rotational inertia, which they do not account for, can partially prevent "memory loss" with increasing rotational diffusion. We describe the two-dimensional motion of a self-propelled particle with both translational and rotational inertia and velocity fluctuations. The proposed generalized analytical equations for the mean kinetic energy, mean-square displacement and noise-averaged trajectory of the self-propelled particle are confirmed by numerical simulations in a wide range of self-propulsion velocities, moments of inertia, rotational diffusivities, medium viscosities and observation times.

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.

Premelting controlled active matter in ice

Self-propelled particles can undergo complex dynamics due to a range of bulk and surface interactions. When a particle is embedded in a host solid near its bulk melting temperature, the latter may melt at the surface of the former in a process known as interfacial premelting. The thickness of the melt film depends on the temperature, impurities, material properties and geometry. A temperature gradient is accompanied by a thermomolecular pressure gradient that drives the interfacial liquid from high to low temperatures and hence the particle from low to high temperatures, in a process called thermal regelation. When the host material is ice and the embedded particle is a biological entity, one has a particularly different form of active matter, which addresses interplay between a wide range of problems, from extremophiles of both terrestrial and exobiological relevance to ecological dynamics in Earth's cryosphere. Of basic importance in all such settings is the combined influence of biological activity and thermal regelation in controlling the redistribution of bioparticles. Therefore, we recast this class of regelation phenomena in the stochastic framework of active Ornstein-Uhlenbeck dynamics and make predictions relevant to this and related problems of interest in biological and geophysical problems. We examine how thermal regelation compromises paleoclimate studies in the context of ice core dating and we find that the activity influences particle dynamics during thermal regelation by enhancing the effective diffusion coefficient. Therefore, accurate dating relies on a quantitative treatment of both effects.

Generalized persistence dynamics for active motion

We analyze the statistical physics of self-propelled particles from a general theoretical framework that properly describes the most salient characteristic of active motion, persistence, in arbitrary spatial dimensions. Such a framework allows the development of a Smoluchowski-like equation for the probability density of finding a particle at a given position and time, without assuming an explicit orientational dynamics of the self-propelling velocity as Langevin-like equation-based models do. Also, the Brownian motion due to thermal fluctuations and the active one due to a general intrinsic persistent motion of the particle are taken into consideration on an equal footing. The persistence of motion is introduced in our formalism in the form of a two-time memory function, K(t,t^{'}). We focus on the consequences when K(t,t^{'})∼(t/t^{'})^{-η}exp[-Γ(t-t^{'})], Γ being the characteristic persistence time, and show that it precisely describes a variety of active motion patterns characterized by η. We find analytical expressions for the experimentally obtainable intermediate scattering function, the time dependence of the mean-squared displacement, and the kurtosis.

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.

Condensation transition in the late-time position of a run-and-tumble particle

We study the position distribution P(R[over ⃗],N) of a run-and-tumble particle (RTP) in arbitrary dimension d, after N runs. We assume that the constant speed v>0 of the particle during each running phase is independently drawn from a probability distribution W(v) and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, P(R[over ⃗],N)→P(R,N) where R=|R[over ⃗]|. We show that, under certain conditions on d and W(v) and for large N, a condensation transition occurs at some critical value of R=R_{c}∼O(N) located in the large-deviation regime of P(R,N). For R<R_{c} (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with R>R_{c} is typically dominated by a "condensate," i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions W(v)=α(1-v/v_{0})^{α-1}/v_{0}, parametrized by α>0, we show that, for large N, P(R,N)∼exp[-Nψ_{d,α}(R/N)], and we compute exactly the rate function ψ_{d,α}(z) for any d and α. We show that the transition manifests itself as a singularity of this rate function at R=R_{c} and that its order depends continuously on d and α. We also compute the distribution of the condensate size for R>R_{c}. Finally, we study the model when the total duration T of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision ∼10^{-100}.

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.

Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems

We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single-particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent of the friction coefficient.

Active Trap Model

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

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.