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Emergent cooperative functionality in active matter systems plays a crucial role in various applications of active swarms, ranging from pollutant foraging and collective threat detection to tissue embolization. In nature, animals like bats and whales use acoustic signals to communicate and enhance their evolutionary competitiveness. Here, we show that information exchange by acoustic waves between active agents creates a large variety of multifunctional structures. In our realization of collective swarms, each unit is equipped with an acoustic emitter and a detector. The swarmers respond to the resulting acoustic field by adjusting their emission frequency and migrating toward the strongest signal. We find self-organized structures with different morphology, including snake-like self-propelled entities, localized aggregates, and spinning rings. These collective swarms exhibit emergent functionalities, such as phenotype robustness, collective decision-making, and environmental sensing. For instance, the collectives show self-regeneration after strong distortion, allowing them to penetrate through narrow constrictions. Additionally, they exhibit a population-scale perception of reflecting objects and a collective response to acoustic control inputs. Our results provide insights into fundamental organization mechanisms in information-exchanging swarms. They may inspire design principles for technical implementations in the form of acoustically or electromagnetically communicating microrobotic swarms capable of performing complex tasks and concerting collective responses to external cues.

Active turbulence, or chaotic self-organized collective motion, is often observed in concentrated suspensions of motile bacteria and other systems of self-propelled interacting agents. To date, there is no fundamental understanding of how geometrical confinement orchestrates active turbulence and alters its physical properties. Here, by combining large-scale experiments, computer modeling, and analytical theory, we have discovered a generic sequence of transitions occurring in bacterial suspensions confined in cylindrical wells of varying radii. With increasing the well's radius, we observed that persistent vortex motion gives way to periodic vortex reversals, four-vortex pulsations, and then well-developed active turbulence. Using computational modeling and analytical theory, we have shown that vortex reversal results from the nonlinear interaction of the first three azimuthal modes that become unstable with the radius increase. The analytical results account for our key experimental findings. To further validate our approach, we reconstructed equations of motion from experimental data. Our findings shed light on the universal properties of confined bacterial active matter and can be applied to various biological and synthetic active systems., Comment: 8 pages, 4 figures in the main text + 15 pages, 7 figures in Supplementary Material

Controlling the swimming behavior of bacteria is crucial, for example, to prevent contamination of ducts and catheters. We show the bacteria modeled by deformable microswimmers can accumulate in flows through straight microchannels either in their center or on previously unknown attractors near the channel walls. We predict a novel resonance effect for semiflexible microswimmers in flows through wavy microchannels. As a result, microswimmers can be deflected in a controlled manner so that they swim in modulated channels distributed over the channel cross-section rather than localized near the wall or the channel center. Thus, depending on the flow amplitude, both upstream orientation of swimmers and their accumulation at the boundaries which can lead to surface rheotaxis are suppressed. Our results suggest new strategies for controlling the behavior of live and synthetic swimmers in microchannels.

Active matter broadly covers the dynamics of self-propelled particles. While the onset of collective behavior in homogenous active systems is relatively well understood, the effect of inhomogeneities such as obstacles and traps lacks overall clarity. Here, we study how interacting self-propelled particles become trapped and released from a trap. We have found that captured particles aggregate into an orbiting condensate with a crystalline structure. As more particles are added, the trapped condensate escape as a whole. Our results shed light on the effects of confinement and quenched disorder in active matter.

Active systems comprised of self-propelled units show fascinating transitions from Brownian-like dynamics to collective coherent motion. Swirling of swimming bacteria is a spectacular example. This study demonstrates that a nematic liquid crystal environment patterned as a spiral vortex controls individual-to-collective transition in bacterial swirls and defines whether they expand or shrink. In dilute dispersions, the bacteria swim along open spiral trajectories, following the pre-imposed molecular orientation. The trajectories are nonpolar. As their concentration exceeds some threshold, the bacteria condense into unipolar circular swirls resembling stable limit cycles. This collective circular motion is controlled by the spiral angle that defines the splay-to-bend ratio of the background director. Vortices with dominating splay shrink the swirls towards the center, while vortices with dominating bend expand them to the periphery. 45o spiraling vortices with splay-bend parity produce the most stable swirls. All the dynamic scenarios are explained by hydrodynamic interactions of bacteria mediated by the patterned passive nematic environment and by the coupling between the concentration and orientation. The acquired knowledge of how to control individual and collective motion of microswimmers by a nematic environment can help in the development of microscopic mechanical systems., Comment: Main text (39 pages and 9 figures) and supplementary materials (10 pages and 6 figures)

Through billions of years of evolution, microorganisms mastered unique swimming behaviors to thrive in complex fluid environments. Limitations in nanofabrication have thus far hindered the ability to design and program synthetic swimmers with the same abilities. Here we encode multi-behavioral responses in artificial swimmers such as microscopic, self-propelled tori using nanoscale 3D printing. We show experimentally and theoretically that the tori continuously transition between two primary swimming modes in response to a magnetic field. The tori also manipulate and transport other artificial swimmers, bimetallic nanorods, as well as passive colloidal particles. In the first behavioral mode, the tori accumulate and transport nanorods; in the second mode, nanorods align along the tori's self-generated streamlines. Our results indicate that such shape-programmed microswimmers have the potential to manipulate biological active matter, e.g. bacteria or cells.

Active matter exhibits remarkable collective behavior in which flows, continuously generated by active particles, are intertwined with the orientational order of these particles. The relationship remains poorly understood as the activity and order are difficult to control independently. Here we demonstrate important facets of this interplay by exploring dynamics of swimming bacteria in a liquid crystalline environment with pre-designed periodic splay and bend in molecular orientation. The bacteria are expelled from the bend regions and condense into polar jets that propagate and transport cargo unidirectionally along the splay regions. The bacterial jets remain stable even when the local concentration exceeds the threshold of bending instability in a non-patterned system. Collective polar propulsion and different role of bend and splay are explained by an advection-diffusion model and by numerical simulations that treat the system as a two-phase active nematic. The ability of prepatterned liquid crystalline medium to streamline the chaotic movements of swimming bacteria into polar jets that can carry cargo along a predesigned trajectory opens the door for potential applications in cell sorting, microscale delivery and soft microrobotics., Comment: Main text (17 pages and 5 figures) and supplementary materials (10 pages and 4 figures)

Active matter research focuses on the emergent behavior among interacting self-propelled particles. Unification of seemingly disconnected paradigms -- active phase-separation of repulsive discs and collective motion of self-propelled rods -- is a major challenge in contemporary active matter. Inspired by the quanto-mechanical wave-particle duality, we develop an approach based on the representation of active particles by smoothed continuum fields. On the basis of the collision kinetics, we demonstrate analytically and numerically how nonequilibrium stresses acting among self-driven, anisotropic objects hinder the formation of phase-separated states as observed for self-propelled discs and facilitate the emergence of orientational order. Besides particle shape, the rigidity of self-propelled objects controlling the symmetry of emergent ordered states is as a crucial parameter: impenetrable, anisotropic rods are found to form polar, moving clusters, whereas large-scale nematic structures emerge for soft rods, notably separated by a bistable coexistence regime. These results indicate that the symmetry of the ordered state is not dictated by the symmetry of the interaction potential but is rather a dynamical, emergent property of active systems. This unifying theoretical framework can represent a variety of active systems: living cell tissues, bacterial colonies, cytoskeletal extracts as well as shaken granular media., Comment: 14 pages, 8 figures

We study dry, dense active nematics at both particle and continuous levels. Specifically, extending the Boltzmann-Ginzburg-Landau approach, we derive well-behaved hydrodynamic equations from a Vicsek-style model with nematic alignment and pairwise repulsion. An extensive study of the phase diagram shows qualitative agreement between the two levels of description. We find in particular that the dynamics of topological defects strongly depends on parameters and can lead to ``arch'' solutions forming a globally polar, smectic arrangement of N\'eel walls. We show how these configurations are at the origin of the defect ordered states reported previously. This work offers a detailed understanding of the theoretical description of dense active nematics directly rooted in their microscopic dynamics.

The motion of Abrikosov vortices in type-II superconductors results in a finite resistance in the presence of an applied electric current. Elimination or reduction of the resistance via immobilization of vortices is the "holy grail" of superconductivity research. Common wisdom dictates that an increase in the magnetic field escalates the loss of energy since the number of vortices increases. Here we show that this is no longer true if the magnetic field and the current are applied parallel to each other.Our experimental studies on the resistive behavior of a superconducting Mo$_{0.79}$Ge$_{0.21}$ nanostrip reveal the emergence of a dissipative state with increasing magnetic field, followed by a pronounced resistance drop, signifying a reentrance to the superconducting state. Large-scale simulations of the 3D time-dependent Ginzburg-Landau model indicate that the intermediate resistive state is due to an unwinding of twisted vortices. When the magnetic field increases, this instability is suppressed due to a better accommodation of the vortex lattice to the pinning configuration. Our findings show that magnetic field and geometrical confinement can suppress the dissipation induced by vortex motion and thus radically improve the performance of superconducting materials., Comment: 10 pages, 4 figures

We consider active particles swimming in a convergent fluid flow in a trapezoid nozzle with no-slip walls. We use mathematical modeling to analyze trajectories of these particles inside the nozzle. By extensive Monte Carlo simulations, we show that trajectories are strongly affected by the background fluid flow and geometry of the nozzle leading to wall accumulation and upstream motion (rheotaxis). In particular, we describe the non-trivial focusing of active rods depending on physical and geometrical parameters. It is also established that the convergent component of the background flow leads to stability of both downstream and upstream swimming at the centerline. The stability of downstream swimming enhances focusing, and the stability of upstream swimming enables rheotaxis in the bulk., Comment: 21 pages, 13 figures

Superconducting vortices and phase slips are primary mechanisms of dissipation in superconducting, superfluid, and cold atom systems. While the dynamics of vortices is fairly well described, phase slips occurring in quasi-one dimensional superconducting wires still elude understanding. The main reason is that phase slips are strongly non-linear time-dependent phenomena that cannot be cast in terms of small perturbations of the superconducting state. Here we study phase slips occurring in superconducting weak links. Thanks to partial suppression of superconductivity in weak links, we employ a weakly nonlinear approximation for dynamic phase slips. This approximation is not valid for homogeneous superconducting wires and slabs. Using the numerical solution of the time-dependent Ginzburg-Landau equation and bifurcation analysis of stationary solutions, we show that the onset of phase slips occurs via an infinite period bifurcation, which is manifested in a specific voltage-current dependence. Our analytical results are in good agreement with simulations., Comment: 13 pages, 11 figures

To survive in harsh conditions, motile bacteria swim in complex environment and respond to the surrounding flow. Here we develop a PDE model describing how the flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease of the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random re-orientation) as it was done previously to explain the viscosity reduction. Second, we demonstrate a possibility of bacterium escape from the wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

We show that memory, in the form of underdamped angular dynamics, is a crucial ingredient for the collective properties of self-propelled particles. Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular velocity, we uncover a rich variety of collective phases not observed in usual overdamped systems, including vortex lattices and active foams. In a model with strictly nematic interactions the smectic arrangement of Vicsek waves giving rise to global polar order is observed. We also provide a calculation of the effective interaction between vortices in the case where a telegraphic noise process is at play, explaining thus the emergence and structure of the vortex lattices observed here and in motility assay experiments., Comment: 5 pages, 4 figures

In the framework of a phase field model of a single cell crawling on a substrate, we investigate how the properties of the cell membrane affect the shape and motility of the cell. Since the membrane influences the cell dynamics on multiple levels and provides a nontrivial feedback, we consider the following fundamental interactions: (i) the reduction of the actin polymerization rate by membrane tension; (ii) area conservation of the cell's two-dimensional cross-section vs. conservation of its circumference (i.e. membrane inextensibility); and (iii) the contribution from the membrane's bending energy to the shape and integrity of the cell. As in experiments, we investigate two pertinent observables -- the cell's velocity and its aspect ratio. We find that the most important effect is the feedback of membrane tension on the actin polymerization. Bending rigidity has only minor effects, visible mostly in dynamic reshaping events, as exemplified by collisions of the cell with an obstacle.

The motion of a pair of micro-wedges ("carriers") in a turbulent bacterial bath is explored using computer simulations with explicit modeling of the bacteria and experiments. The orientation of the two micro-wedges is parallel and fixed by an applied magnetic field but the translational coordinates can move freely as induced by the bacterial bath. As a result, two carriers of same orientation move such that their mutual distance decreases. Eventually the two carriers stack on each other with no intervening bacteria exhibiting a stable dynamical mode where the two micro-wedges follow each other with the same velocity. These findings are in qualitative agreement with experiments on two micro-wedges in a bacterial bath. Our results provide insight into understanding the self-assembly of many micro-wedges in an active bath., Comment: 11 pages, 6 figures

We investigate equilibration processes shortly after sudden perturbations are applied to ultracold trapped superfluids. We show the similarity of phase imprinting and localized density depletion perturbations, both of which initially are found to produce "phase walls". These planar defects are associated with a sharp gradient in the phase. Importantly they relax following a quite general sequence. Our studies, based on simulations of the complex time-dependent Ginzburg-Landau equation, address the challenge posed by these experiments: how a superfluid eventually eliminates a spatially extended planar defect. The processes involved are necessarily more complex than equilibration involving simpler line vortices. An essential mechanism for relaxation involves repeated formation and loss of vortex rings near the trap edge., Comment: Expanded version: 11 pages, 12 figures

Recently, it was found that a wedgelike microparticle (referred to as "carrier") which is only allowed to translate but not to rotate exhibits a directed translational motion along the wedge cusp if it is exposed to a bath of microswimmers. Here we model this effect in detail by resolving the microswimmers explicitly using interaction models with different degrees of mutual alignment. Using computer simulations we study the impact of these interactions on the transport efficiency of V-shaped carrier. We show that the transport mechanisms itself strongly depends on the degree of alignment embodied in the modelling of the individual swimmer dynamics. For weak alignment, optimal carrier transport occurs in the turbulent microswimmer state and is induced by swirl depletion inside the carrier. For strong aligning interactions, optimal transport occurs already in the dilute regime and is mediated by a polar cloud of swimmers in the carrier wake pushing the wedge-particle forward. We also demonstrate that the optimal shape of the carrier leading to maximal transport speed depends on the kind of interaction model used., Comment: 7 pages, 7 figures

We demonstrate that collective turbulent-like motion in a bacterial bath can power and steer directed transport of mesoscopic carriers through the suspension. In our experiments and simulations, a microwedge-like "bulldozer" draws energy from a bacterial bath of varied density. We obtain that a maximal transport speed is achieved in the turbulent state of the bacterial suspension. This apparent rectification of random motion of bacteria is caused by polar ordered bacteria inside the cusp region of the carrier, which is shielded from the outside turbulent fluctuations., Comment: 5 pages, 5 figures

We present numerical simulations of phase imprinting experiments in ultracold trapped Fermi gases which are in good agreement with recent, independent experimental results. Our focus is on the sequence and evolution of defects using the fermionic time-dependent Ginzburg-Landau equation, which contains dissipation necessary for equilibration. In contrast to other simulations we introduce small, experimentally unavoidable symmetry breaking, particularly that associated with thermal fluctuations and with the phase imprinting tilt angle, and illustrate their dramatic effects. The former causes vortex rings in confined geometries to move to the trap surface and rapidly decay into more stable vortex lines, as appears consistent with recent experimental claims. The latter aligns the precessing and relatively long-lived vortex filaments, rendering them difficult to distinguish from solitons., Comment: 5 pages, 2 figures, with included supplementary material

Collective motion of self-propelled organisms or synthetic particles often termed active fluid has attracted enormous attention in broad scientific community because of it fundamentally non-equilibrium nature. Energy input and interactions among the moving units and the medium lead to complex dynamics. Here we introduce a new class of active matter, living liquid crystals (LLCs) that combine living swimming bacteria with a lyotropic liquid crystal. The physical properties of LLCs can be controlled by the amount of oxygen available to bacteria, by concentration of ingredients, or by temperature. Our studies reveal a wealth of new intriguing dynamic phenomena, caused by the coupling between the activity-triggered flow and long-range orientational order of the medium. Among these are (a) non-linear trajectories of bacterial motion guided by non-uniform director, (b) local melting of the liquid crystal caused by the bacteria-produced shear flows, (c) activity-triggered transition from a non-flowing uniform state into a flowing one-dimensional periodic pattern and its evolution into a turbulent array of topological defects, (d) birefringence-enabled visualization of microflow generated by the nanometers-thick bacterial flagella. Unlike their isotropic counterpart, the LLCs show collective dynamic effects at very low volume fraction of bacteria, on the order of 0.2%. Our work suggests an unorthodox design concept to control and manipulate the dynamic behavior of soft active matter and opens the door for potential biosensing and biomedical applications., Comment: 32 pages, 8 figures, Supporting Information included

We show that "dry" active nematics, e.g. collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, band-like structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of the relatively simple and well-known (deterministic) hydrodynamic equations describing these systems in a dilute limit, and of a self-propelled particle Vicsek-like model for this class of active matter. In this last case, revisiting the status of the strong fluctuations and long-range correlations now considered as landmarks of orientationally-ordered active phases, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous density fluctuations are present in the homogeneous quasi-ordered nematic phase and characterized by a non-trivial scaling exponent.

We derive a set of minimal yet complete nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for the fronts forming density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Self-organized regular patterns are ubiquitous in nature, and one of their most celebrated manifestations is the Abrikosov vortex lattice: under an applied magnetic field, the homogeneous superconductivity becomes unstable and cast itself into a regular texture of the "normal" filaments, called Abrikosov vortices, immersed into a superconducting matrix. Its prediction and the experimental discovery became a breakthrough in our understanding of superconductivity and founded a new direction in physics. Here we show that the interplay between the superconducting order parameter and elastic fields, which are intimately connected to the very existence of the superconductivity itself, can result in a novel superconducting state dual to the Abrikosov state: a regular texture of superconducting islands. The fact that both patterns emerge within the framework of the Ginzburg-Landau description of superconductivity indicates that the formation of regular structures may be a generic feature of any phase transition. Emergence of superconducting island arrays is not specific to the effect of the elastic forces, but can be caused by any inherent mechanism generating long-range non-local interactions in the Ginzburg-Landau functional, for example, by the Coulomb forces. In particular, our findings suggest the formation of a superconducting island textures as a scenario for a superconductor-to-insulator transition in thin films., Comment: 9 pages, 5 figures, revtex4

We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by pushing" or "pulling" both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which, complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria., Comment: 28 pages, 20 figures

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions., Comment: 15 pages, 3 figures, submitted to Physical Biology

The influence of controlled vibrations on the granular rheology is investigated in a specifically designed experiment. We study experimentally a thin granular film spreading under the action of horizontal vibrations. A nonlinear diffusion equation is derived theoretically that describes the evolution of the deposit shape. A self-similar parabolic shape (the``granular droplet'') and a spreading dynamics are predicted that both agree quantitatively with the experimental results. The theoretical analysis is used to extract effective dynamic friction coefficients between the base and the granular layer under sustained and controlled vibrations. We derive an empirical friction law involving external driving parameters and the layer height to represent our data., Comment: 4 pages, 3 figures

Crosslinks and molecular motors play an important role in the organization of cytoskeletal filament networks. Here we incorporate the effect of crosslinks into our model of polar motor-filament organization [Phys. Rev. E {\bf 71}, 050901 (2005)], through suppressing the relative sliding of filaments in the course of motor-mediated alignment. We show that this modification leads to a nontrivial macroscopic behavior, namely the oriented state exhibits a transverse instability in contrast to the isotropic instability that occurs without crosslinks. This transverse instability leads to the formation of dense extended bundles of oriented filaments, similar to recently observed structures in actomyosin. This model also can be applied to situations with two oppositely directed motor species or motors with different processing speeds., Comment: 15 pages, 5 figures, submitted to New. J. Phys

Avalanche experiments on an erodible substrate are treated in the framework of ``partial fluidization'' model of dense granular flows. The model identifies a family of propagating soliton-like avalanches with shape and velocity controlled by the inclination angle and the depth of substrate. At high inclination angles the solitons display a transverse instability, followed by coarsening and fingering similar to recent experimental observation. A primary cause for the transverse instability is directly related to the dependence of soliton velocity on the granular mass trapped in the avalanche., Comment: 3 figures, 4 pages, submitted to Phys Rev Lett

Granular materials are ubiquitous in our daily lives. While they have been a subject of intensive engineering research for centuries, in the last decade granular matter attracted significant attention of physicists. Yet despite a major efforts by many groups, the theoretical description of granular systems remains largely a plethora of different, often contradicting concepts and approaches. Authors give an overview of various theoretical models emerged in the physics of granular matter, with the focus on the onset of collective behavior and pattern formation. Their aim is two-fold: to identify general principles common for granular systems and other complex non-equilibrium systems, and to elucidate important distinctions between collective behavior in granular and continuum pattern-forming systems., Comment: Submitted to Reviews of Modern Physics. Full text with figures (2Mb pdf) avaliable at http://mti.msd.anl.gov/AransonTsimringReview/aranson_tsimring.pdf Community responce is appreciated. Comments/suggestions send to aronson@msd.anl.gov

We derive a model describing spatio-temporal organization of an array of microtubules interacting via molecular motors. Starting from a stochastic model of inelastic polar rods with a generic anisotropic interaction kernel we obtain a set of equations for the local rods concentration and orientation. At large enough mean density of rods and concentration of motors, the model describes orientational instability. We demonstrate that the orientational instability leads to the formation of vortices and (for large density and/or kernel anisotropy) asters seen in recent experiments., Comment: 4 pages, 5 figures, to appear in Phys. Rev. E, Rapid Communications

This paper has been withdrawn due to a request of the first author to exclude his name from the author list and to motivate sufficiently the appliclication of the used model for antiferromagnetic insulators., Comment: This paper has been withdrawn

We present numerical and analytical studies of coupled nonlinear Maxwell and thermal diffusion equations which describe nonisothermal dendritic flux penetration in superconducting films. We show that spontaneous branching of propagating flux filaments occurs due to nonlocal magnetic flux diffusion and positive feedback between flux motion and Joule heat generation. The branching is triggered by a thermomagnetic edge instability which causes stratification of the critical state. The resulting distribution of magnetic microavalanches depends on a spatial distribution of defects. Our results are in good agreement with experiments performed on Nb films., Comment: 4 pages, 3 figures, see http://mti.msd.anl.gov/aran_h1.htm for extensive collection of movies of dendritic flux and temperature patterns

We carry out a detailed comparison of soft particle molecular dynamics simulations with the theory of partially fluidized shear granular flows. We verify by direct simulations a constitutive relation based on the separation of the shear stress tensor into a fluid part proportional to the strain rate tensor, and a remaining solid part. The ratio of these two components is determined by the order parameter. Based on results of the simulations we construct the ``free energy'' function for the order parameter. We also present the simulations of the stationary deep 2D granular flows driven by an upper wall and compare it with the continuum theory., Comment: 4 pages, 4 figures, submitted to PRL