Nonlinear Response of Inertial Tracers in Steady Laminar Flows: Differential and Absolute Negative Mobility

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries

Strong, scale-free disorder disrupts typical transport properties like the Stokes-Einstein relation and linear response, leading to anomalous diffusion observed in amorphous materials, glasses, living cells, and other systems. Our study reveals that the combination of scale-free quenched disorder and geometrical constraints induces unconventional single-particle mobility behavior. Specifically, in a two-dimensional channel with width w, under external drive, tighter geometrical constraints (smaller w) enhance mobility. We derive an explicit form of the response to an external force by utilizing the double-subordination approach for the quenched trap model. The observed mobility enhancement occurs in the low-temperature regime where the distribution of localization times is scale-free.

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.

Diffusion of noiseless active particles in a planar convection array

We investigated, both analytically and numerically, the dynamics of a noiseless overdamped active particle in a square lattice of planar counter-rotating convection rolls. Below a first threshold of the self-propulsion speed, a fraction of the simulated particle's trajectories spatially diffuse around the convection rolls, whereas the remaining trajectories remain trapped inside the injection roll. We detected two chaotic diffusion regimes: (i) below a second, higher threshold of the self-propulsion speed, the particle performs a random motion characterized by asymptotic normal diffusion. Long superdiffusive transients were observed for vanishing small self-propulsion speeds. (ii) above that threshold, the particle follows chaotic running trajectories with speed and orientation close to those of the self-propulsion vector at injection and its dynamics is superdiffusive. Chaotic diffusion disappears in the ballistic limit of extremely large self-propulsion speeds.

Paradoxical nature of negative mobility in the weak dissipation regime

We reinvestigate a paradigmatic model of nonequilibrium statistical physics consisting of an inertial Brownian particle in a symmetric periodic potential subjected to both a time-periodic force and a static bias. In doing so, we focus on the negative mobility phenomenon in which the average velocity of the particle is opposite to the constant force acting on it. Surprisingly, we find that in the weak dissipation regime, thermal fluctuations induce negative mobility much more frequently than it happens if dissipation is stronger. In particular, for the very first time, we report a parameter set in which thermal noise causes this effect in the nonlinear response regime. Moreover, we show that the coexistence of deterministic negative mobility and chaos is routinely encountered when approaching the overdamped limit in which chaos does not emerge rather than near the Hamiltonian regime of which chaos is one of the hallmarks. On the other hand, at non-zero temperature, the negative mobility in the weak dissipation regime is typically affected by weak ergodicity breaking. Our findings can be corroborated experimentally in a multitude of physical realizations, including, e.g., Josephson junctions and cold atoms dwelling in optical lattices.

Absolute Negative Mobility of an Active Tracer in a Crowded Environment

Absolute negative mobility (ANM) refers to the situation where the average velocity of a driven tracer is opposite to the direction of the driving force. This effect was evidenced in different models of nonequilibrium transport in complex environments, whose description remains effective. Here, we provide a microscopic theory for this phenomenon. We show that it emerges in the model of an active tracer particle submitted to an external force and which evolves on a discrete lattice populated with mobile passive crowders. Resorting to a decoupling approximation, we compute analytically the velocity of the tracer particle as a function of the different parameters of the system and confront our results to numerical simulations. We determine the range of parameters where ANM can be observed, characterize the response of the environment to the displacement of the tracer, and clarify the mechanism underlying ANM and its relationship with negative differential mobility (another hallmark of driven systems far from the linear response).

Macroscopic charge segregation in driven polyelectrolyte solutions

Understanding the behavior of charged complex fluids is crucial for a plethora of important industrial, technological, and medical applications. Here, using coarse-grained molecular dynamics simulations, we investigate the properties of a polyelectrolyte solution with explicit counterions and implicit solvent that is driven by a steady electric field. By properly tuning the interplay between interparticle electrostatics and the applied electric field, we uncover two non-equilibrium continuous phase transitions as a function of the driving field. The first transition occurs from a homogeneous mixed phase to a macroscopic charge-segregated phase in which the polyelectrolyte solution self-organizes to form two lanes of like-charges, parallel to the applied field. We show that the fundamental underlying factor responsible for the emergence of this charge segregation in the presence of an electric field is the excluded volume interactions of the drifting polyelectrolyte chains. As the driving field is increased further, a re-entrant transition is observed from a charge-segregated phase to a homogeneous phase. The re-entrance is signaled by a decrease in the mobility of the monomers and counterions as the electric field is increased. Furthermore, with multivalent counterions, a counterintuitive regime of negative differential mobility is observed in which the charges move progressively more slowly as the driving field is increased. We show that all these features can be consistently explained using an intuitive trapping mechanism that operates between the oppositely moving charges, and present numerical evidence to support our claims. Parameter dependencies and phase diagrams are studied to better understand charge segregation in such driven polyelectrolyte solutions.

Absolute negative mobility of active polymer chains in steady laminar flows

We investigate the transport of active polymer chains in steady laminar flows in the presence of thermal noise and an external constant force. In the model, the polymer chain is worm-like and is propelled by active forces along its tangent vectors. Compared with inertial Brownian particles, active polymer chains in steady laminar flows exhibit richer movement patterns due to their specific spatial structures. The simulation results show that the velocity-force relation is strongly dependent on the system parameters such as the chain length, bending rigidity, active force and so on. The polymer chain may move in some preferential movement directions and exhibits absolute negative mobility within appropriate parameter regimes, i.e., the polymer chain can move in a direction opposite to the external constant force. In particular, we can observe giant negative mobility in a broad range of parameter regimes.

Negative rectification and anomalous diffusion in nonlinear substrate potentials: Dynamical relaxation and information entropy

We numerically investigate the rectification of the probability flux and dynamical relaxation of particles moving in a system with and without noise. The system, driven by two external forces, consists of two substrate potentials that have identical shapes and different potential barriers with different friction coefficients. The deterministic model exhibits the perfect rectification of the probability flux, ratchet effect, and the dependence of the unpredictability of the dynamics on basin of attraction. In contrast, the stochastic model displays that the rectification is sensitive to the temperature and an external bias. They can induce kinetic phase transitions between no transport and a finite net transport. These transitions lead to an unexpected phenomenon, called negative rectification. The results are analyzed through the corresponding time-dependent diffusion coefficient, information entropy (IE), etc. At a low temperature, anomalous diffusions occur in system. For the occurrence of the flux in certain parameter regimes, the larger the diffusion is, the smaller the corresponding IE is, and vice versa. We also present the selected parameter regimes for the emergence of the rectification and negative rectification. Additionally, we study the rectification of the interacting particles in the system and find that the flux may depend on the coupling strength and the number of the interacting particles, and that collective motions occur for the forward flux. Our work provides not only a way of the rectification for the transport of various particles (e.g., ions, electrons, photons, phonons, molecules, DNA chains, nanoswimmers, dust particles, etc.) in physics, chemistry, biology, and material science, but also a design of various circuits.

Origin of dispersionless transport in spite of thermal noise

The "dispersionless transport" of a weakly damped Brownian particle in a tilted periodic potential is defined by (i) a plateau of the particle's coordinate dispersion extending over a very broad time interval and (ii) by the impossibility to measure the diffusion coefficient within this plateau region. While the first part of this definition has been explained in the literature, the second part has been thought to follow from (i). Here, the impossibility to measure the diffusion coefficient is shown to be actually due to the wild fluctuations of the dispersion itself in the plateau region. An expression for the timescale over which a reliable determination of the diffusion coefficient is possible is derived. A procedure that allows accurate determination of the diffusion coefficient by observing the particle trajectory only within a small part of the plateau region is suggested and shown to be feasible by numerical simulations of a weakly damped Brownian particle in a tilted washboard potential.

Inverse currents in Coulomb-coupled quantum dots

The inverse current, i.e., one induced current is opposite to both applied forces, has recently been found in a classical one-dimensional interacting Hamiltonian system [Phys. Rev. Lett. 124, 110607 (2020)10.1103/PhysRevLett.124.110607]. In this paper, we show that the inverse current also exists in quantum system. Based on Coulomb-coupled quantum dots system, we find that the inverse current will appear when Coulomb interaction increases. This does not violate the second law of thermodynamics, since entropy reduction caused by the inverse current is compensated by entropy increase caused by the forward current, which ensures that total entropy increase of the system is always greater than zero.

Dynamic mode decomposition of inertial particle caustics in Taylor-Green flow

Inertial particles advected by a background flow can show complex structures. We consider inertial particles in a 2D Taylor-Green (TG) flow and characterize particle dynamics as a function of the particle's Stokes number using dynamic mode decomposition (DMD) method from particle image velocimetry (PIV) like-data. We observe the formation of caustic structures and analyze them using DMD to (a) determine the Stokes number of the particles, and (b) estimate the particle Stokes number composition. Our analysis in this idealized flow will provide useful insight to analyze inertial particles in more complex or turbulent flows. We propose that the DMD technique can be used to perform similar analysis on an experimental system.

Anisotropic Diffusion in Driven Convection Arrays

We numerically investigate the transport of a Brownian colloidal particle in a square array of planar counter-rotating convection rolls at high Péclet numbers. We show that an external force produces huge excess peaks of the particle's diffusion constant with a height that depends on the force orientation and intensity. In sharp contrast, the particle's mobility is isotropic and force independent. We relate such a nonlinear response of the system to the advection properties of the laminar flow in the suspension fluid.

Advection-enhanced diffusion in biased convection arrays

We numerically investigated the transport of a passive colloidal particle in a one-dimensional periodic array of planar counter-rotating convection rolls at high Péclet numbers. We show that advection-enhanced diffusion is drastically suppressed by an external transverse bias but strongly reinforced by a longitudinal drive of appropriate intensity. Both effects are magnified by imposing free-slip flows at the array's edges. The dependence of the diffusion constant on an external forcing is interpreted as a measure of the fluid-mechanical robustness of the flow boundary layer mechanism governing diffusion in convection rolls.

Fluctuating Motion in an Active Environment

We derive the fluctuation dynamics of a probe in weak coupling with a living medium, modeled as particles undergoing an active Ornstein-Uhlenbeck dynamics. Nondissipative corrections to the fluctuation-dissipation relation are written out explicitly in terms of time correlations in the active medium. A first term changes the inertial mass of the probe as a consequence of the persistence of the active medium. A second correction modifies the friction kernel. The resulting generalized Langevin equation benchmarks the motion induced on probes immersed in active versus passive media. The derivation uses nonequilibrium response theory.

Diffusion properties of self-propelled particles in cellular flows

We study the dynamics of a self-propelled particle advected by a steady laminar flow. The persistent motion of the self-propelled particle is described by an active Ornstein-Uhlenbeck process. We focus on the diffusivity properties of the particle as a function of persistence time and free-diffusion coefficient, revealing non-monotonic behaviors, with the occurrence of a minimum and a steep growth in the regime of large persistence time. In the latter limit, we obtain an analytical prediction for the scaling of the diffusion coefficient with the parameters of the active force. Our study sheds light on the effect of a flow-field on the diffusion of active particles, such as living microorganisms and motile phytoplankton in fluids.

Negative mobility, sliding, and delocalization for stochastic networks

We consider prototype configurations for quasi-one-dimensional stochastic networks that exhibit negative mobility, meaning that current decreases or even reversed as the bias is increased. We then explore the implications of disorder. In particular, we ask whether lower and upper bias thresholds restrict the possibility to witness nonzero current (sliding and antisliding transitions, respectively), and whether a delocalization effect manifests itself (crossover from over-damped to under-damped relaxation). In the latter context detailed analysis of the relaxation spectrum as a function of the bias is provided for both on-chain and off-chain disorder.

Inverse Currents in Hamiltonian Coupled Transport

The occurrence of an inverse current, where the sign of the induced current is opposite to the applied force, is a highly counterintuitive phenomenon. We show that inverse currents in coupled transport (ICC) of energy and particle can occur in a one-dimensional interacting Hamiltonian system when its equilibrium state is perturbed by coupled thermodynamic forces. This seemingly paradoxical result is possible due to the self-organization occurring in the system in response to the applied forces.

Spontaneous rectification and absolute negative mobility of inertial Brownian particles induced by Gaussian potentials in steady laminar flows

We study the transport of inertial Brownian particles in steady laminar flows in the presence of two-dimensional Gaussian potentials. Through extensive numerical simulations, it is found that the transport is sensitively dependent on the external constant force and the Gaussian potential. Within tailored parameter regimes, the system exhibits a rich variety of transport behaviors. There exists the phenomenon of spontaneous rectification (SR), where the directed transport of particles can occur in the absence of any external driving forces. It is found that SR of the particles can be manipulated by the spatial position of the Gaussian potential. Moreover, when the potential lies at the center of the cellular flow, the system exhibits absolute negative mobility (ANM), i.e., the particles can move in a direction opposite to the constant force. More importantly, the phenomenon of ANM induced by Gaussian potentials is robust in a wide range of system parameters and can be further strengthened with the optimized parameters, which may pave the way to the implementation of related experiments.

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Two-dimensional chaotic thermostat and behavior of a thermalized charge in a weak magnetic field

A two-dimensional version of a chaotic thermostat is investigated. Its structure follows the concept previously introduced by the author [G. J. Morales, Phys. Rev. E 97, 032203 (2018)2470-004510.1103/PhysRevE.97.032203] to generate a one-dimensional chaotic thermostat, namely, the usual friction force of a deterministic thermostat is supplemented with a self-consistent fluctuating force that depends on the drag coefficient associated with coupling to the heat bath. Azimuthal symmetry requires the thermostat to have two internal degrees of freedom, thus the Martyna-Klein-Tuckerman [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)JCPSA60021-960610.1063/1.463940] model is chosen for the heat bath. The unmagnetized system exhibits two-dimensional diffusive behavior, achieves symmetric Maxwellian velocity distributions in the absence of an external potential, and satisfies the Einstein relation when an external force is applied. The velocity fluctuations display the characteristic exponential frequency spectrum associated with chaotic systems. The model is used to explore the diffusive motion of a thermalized charge in a weak magnetic field and the associated Hall and Pedersen mobilities. Over a range of magnetic field strengths the charge exhibits absolute negative mobility.

Nonlinear transport coefficients from large deviation functions

Nonlinear response occurs naturally when a strong perturbation takes a system far from equilibrium. Despite its omnipresence in nanoscale systems, it is difficult to predict in a general and efficient way. Here, we introduce a way to compute arbitrarily high order transport coefficients of stochastic systems, using the framework of large deviation theory. Leveraging time reversibility in the microscopic dynamics, we relate nonlinear response to equilibrium multitime correlation functions among both time reversal symmetric and asymmetric observables, which can be evaluated from derivatives of large deviation functions. This connection establishes a thermodynamiclike relation for nonequilibrium response and provides a practical route to its evaluation, as large deviation functions are amenable to importance sampling. We demonstrate the generality and efficiency of this method in predicting transport coefficients in single particle systems and an interacting system exhibiting thermal rectification.

Tracer diffusion in crowded narrow channels

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Anomalous mobility of a driven active particle in a steady laminar flow

We study, via extensive numerical simulations, the force-velocity curve of an active particle advected by a steady laminar flow, in the nonlinear response regime. Our model for an active particle relies on a colored noise term that mimics its persistent motion over a time scale [Formula: see text]. We find that the active particle dynamics shows non-trivial effects, such as negative differential and absolute mobility (NDM and ANM, respectively). We explore the space of the model parameters and compare the observed behaviors with those obtained for a passive particle ([Formula: see text]) advected by the same laminar flow. Our results show that the phenomena of NDM and ANM are quite robust with respect to the details of the considered noise: in particular for finite [Formula: see text] a more complex force-velocity relation can be observed.

Negative differential mobility in interacting particle systems

Driven particles in the presence of crowded environment, obstacles, or kinetic constraints often exhibit negative differential mobility (NDM) due to their decreased dynamical activity. Based on the empirical studies of conserved lattice gas model, two species exclusion model and other interacting particle systems we propose a new mechanism for complex many-particle systems where slowing down of certain non-driven degrees of freedom by the external field can give rise to NDM. To prove that the slowing down of the non-driven degrees is indeed the underlying cause, we consider several driven diffusive systems including two species exclusion models, misanthrope process, and show from the exact steady state results that NDM indeed appears when some non-driven modes are slowed down deliberately. For clarity, we also provide a simple pedagogical example of two interacting random walkers on a ring which conforms to the proposed scenario.

Transport and diffusion of paramagnetic ellipsoidal particles in a rotating magnetic field

Transport and diffusion of paramagnetic ellipsoidal particles under the action of a rotating magnetic field are numerically investigated in a two-dimensional channel. It is found that paramagnetic ellipsoidal particles in a rotating magnetic field can be rectified in the upper-lower asymmetric channel. The transport and the effective diffusion coefficient are much more different and complicated for active particles, while they have similar behaviors and change a little when applying rotating magnetic fields of different frequencies for passive particles. For active particles, the back-and-forth rotational motion facilitates the effective diffusion coefficient and reduces the rectification, whereas the rotational motion synchronous with the magnetic field suppresses the effective diffusion coefficient and enhances the rectification. There exist optimized values of the parameters (the anisotropic degree, the amplitude and frequency of magnetic field, the self-propelled velocity, and the rotational diffusion rate) at which the average velocity and diffusion take their maximal values. Particles with different shapes, self-propelled speeds, or rotational diffusion rates will move to the opposite directions and can be separated by applying rotating magnetic fields of suitable strength and frequency. Our results can be used to separate particles, orient the particles along any direction at will during motion, and control the particle diffusion.

Response to a small external force and fluctuations of a passive particle in a one-dimensional diffusive environment

We investigate the long-time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant D. We consider two cases: (a) The particle is pulled forward by a small external constant force and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large D limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the crossover to the true asymptotic behavior.

Negative differential mobility and trapping in active matter systems

Using simulations, we examine the average velocity as a function of applied drift force for active matter particles moving through a random obstacle array. We find that for low drift force, there is an initial flow regime where the mobility increases linearly with drive, while for higher drift forces a regime of negative differential mobility appears in which the velocity decreases with increasing drive due to the trapping of active particles behind obstacles. A fully clogged regime exists at very high drift forces when all the particles are permanently trapped behind obstacles. We find for increasing activity that the overall mobility is nonmonotonic, with an enhancement of the mobility for small levels of activity and a decrease in mobility for large activity levels. We show how these effects evolve as a function of disk and obstacle density, active run length, drift force, and motor force.

Finding efficient swimming strategies in a three-dimensional chaotic flow by reinforcement learning

We apply a reinforcement learning algorithm to show how smart particles can learn approximately optimal strategies to navigate in complex flows. In this paper we consider microswimmers in a paradigmatic three-dimensional case given by a stationary superposition of two Arnold-Beltrami-Childress flows with chaotic advection along streamlines. In such a flow, we study the evolution of point-like particles which can decide in which direction to swim, while keeping the velocity amplitude constant. We show that it is sufficient to endow the swimmers with a very restricted set of actions (six fixed swimming directions in our case) to have enough freedom to find efficient strategies to move upward and escape local fluid traps. The key ingredient is the learning-from-experience structure of the algorithm, which assigns positive or negative rewards depending on whether the taken action is, or is not, profitable for the predetermined goal in the long-term horizon. This is another example supporting the efficiency of the reinforcement learning approach to learn how to accomplish difficult tasks in complex fluid environments.

Anomalous force-velocity relation of driven inertial tracers in steady laminar flows

We study the nonlinear response to an external force of an inertial tracer advected by a two-dimensional incompressible laminar flow and subject to thermal noise. In addition to the driving external field F, the main parameters in the system are the noise amplitude [Formula: see text] and the characteristic Stokes time [Formula: see text] of the tracer. The relation velocity vs. force shows interesting effects, such as negative differential mobility (NDM), namely a non-monotonic behavior of the tracer velocity as a function of the applied force, and absolute negative mobility (ANM), i.e. a net motion against the bias. By extensive numerical simulations, we investigate the phase chart in the parameter space of the model, [Formula: see text], identifying the regions where NDM, ANM and more common monotonic behaviors of the force-velocity curve are observed.

Flow Navigation by Smart Microswimmers via Reinforcement Learning

Smart active particles can acquire some limited knowledge of the fluid environment from simple mechanical cues and exert a control on their preferred steering direction. Their goal is to learn the best way to navigate by exploiting the underlying flow whenever possible. As an example, we focus our attention on smart gravitactic swimmers. These are active particles whose task is to reach the highest altitude within some time horizon, given the constraints enforced by fluid mechanics. By means of numerical experiments, we show that swimmers indeed learn nearly optimal strategies just by experience. A reinforcement learning algorithm allows particles to learn effective strategies even in difficult situations when, in the absence of control, they would end up being trapped by flow structures. These strategies are highly nontrivial and cannot be easily guessed in advance. This Letter illustrates the potential of reinforcement learning algorithms to model adaptive behavior in complex flows and paves the way towards the engineering of smart microswimmers that solve difficult navigation problems.