Differently from passive Brownian particles, active particles, also known as self-propelled Brownian particles or microswimmers and nanoswimmers, are capable of taking up energy from their environment and converting it into directed motion. Because of this constant flow of energy, their behavior can be explained and understood only within the framework of nonequilibrium physics. In the biological realm, many cells perform directed motion, for example, as a way to browse for nutrients or to avoid toxins. Inspired by these motile microorganisms, researchers have been developing artificial particles that feature similar swimming behaviors based on different mechanisms. These man-made micromachines and nanomachines hold a great potential as autonomous agents for health care, sustainability, and security applications. With a focus on the basic physical features of the interactions of self-propelled Brownian particles with a crowded and complex environment, this comprehensive review will provide a guided tour through its basic principles, the development of artificial self-propelling microparticles and nanoparticles, and their application to the study of nonequilibrium phenomena, as well as the open challenges that the field is currently facing.

The new dynamic phase diagram for driven vortices with varying lattice softness we present here indicates that, at high driving currents, at least two distinct dynamic phases of flux flow appear depending on the vortex-vortex interaction strength. When the flux lattice is soft, the vortices flow in independently moving channels with smectic structure. For stiff flux lattices, adjacent channels become locked together, producing crystalline-like order in a coupled channel phase. At the crossover lattice softness between these phases, the system produces a maximum amount of voltage noise. Our results relate spatial order with transport and are in agreement with experiments.PACS numbers: 74.60.Ge Nonequilibrium problems involving elastic lattices and disordered media, such as the nature of depinning transitions or the behavior of rapidly driven lattices, appear in a wide variety of systems including superconducting vortex lattices, charge-density waves, and solid-on-solid friction. Much recent interest has been devoted to the motion of a vortex lattice (VL) across a disordered substrate under applied currents both at and well beyond depinning. The reordering of a rapidly-driven VL is supported by simulations [1][2][3][4][5] as well as neutron scattering [6] and decoration experiments [7], but the nature of the order that appears remains a subject of debate. In addition, the relationship of the voltage noise observed near depinning to the microscopic vortex motion at higher currents has not been addressed.Recent work on the behavior of a VL driven by a large current produced conflicting pictures of the VL order, ranging from a crystalline structure [2] to a moving smectic state [8]. In Ref.[9], the VL does not fully recrystallize, but instead enters a moving Bragg glass state with channels. Moreover, Ref. [8] describes the VL in terms of independently moving channels of vortices with overall smectic order. Other theories also focus on channels of vortices [10]. Smectic structure factors (S(q)) of the VL were observed in simulations of vortices moving over strong pinning [3,4], in agreement with Ref. [8].Very recent decoration experiments [12] produced both crystalline-like and smectic order of the moving VL, depending on the magnitude of the applied magnetic field. Smectic order appears at low fields, when the vortices interact weakly, and crystalline-like peaks in S(q) appear at high fields, when the vortex interactions are stronger. This suggests that the softness of the VL is important in determining the vortex behavior at high driving currents.In this paper, we propose a new dynamic phase diagram in which both smectic and crystalline-like order appear as the VL softness is varied. Using simulations of current-driven vortices, we clearly define regions of the phase diagram based on S(q), V (I) curves, voltage noise, velocity distributions, direct observation of the lattice, and defect density calculations. We compute experimentally relevant voltage noise spectra [11] at all currents from depinning to high drives,...

Abstract. We review the depinning and nonequilibrium phases of collectively interacting particle systems driven over random or periodic substrates. This type of system is relevant to vortices in type-II superconductors, sliding charge density waves, electron crystals, colloids, stripe and pattern forming systems, and skyrmions, and could also have connections to jamming, glassy behaviors, and active matter. These systems are also ideal for exploring the broader issues of characterizing transient and steady state nonequilibrium flow phases as well as nonequilibrium phase transitions between distinct dynamical phases, analogous to phase transitions between different equilibrium states. We discuss the differences between elastic and plastic depinning on random substrates and the different types of nonequilibrium phases which are associated with specific features in the velocity-force curves, fluctuation spectra, scaling relations, and local or global particle ordering. We describe how these quantities can change depending on the dimension, anisotropy, disorder strength, and the presence of hysteresis. Within the moving phase we discuss how there can be a transition from a liquid-like state to dynamically ordered moving crystal, smectic, or nematic states. Systems with periodic or quasiperiodic substrates can have multiple nonequilibrium second or first order transitions in the moving state between chaotic and coherent phases, and can exhibit hysteresis. We also discuss systems with competing repulsive and attractive interactions, which undergo dynamical transitions into stripes and other complex morphologies when driven over random substrates. Throughout this work we highlight open issues and future directions such as absorbing phase transitions, nonequilibrium work relations, inertia, the role of non-dissipative dynamics such as Magnus effects, and how these results could be extended to the broader issues of plasticity in crystals, amorphous solids, and jamming phenomena.

As a function of applied field, we find a rich variety of ordered and partially-ordered vortex lattice configurations in systems with square or triangular arrays of pinning sites. We present formulas that predict the matching fields at which commensurate vortex configurations occur and the vortex lattice orientation with respect to the pinning lattice. Our results are in excellent agreement with recent imaging experiments on square pinning arrays [K. Harada et al., Science 274, 1167].

We use particle-based simulations to examine the static and driven collective phases of skyrmions interacting with random quenched disorder. We show that non-dissipative effects due to the Magnus term reduce the depinning threshold and strongly affect the skyrmion motion and the nature of the dynamic phases. The quenched disorder causes the Hall angle to become drive-dependent in the moving skyrmion phase, while different flow regimes produce distinct signatures in the transport curves. For weak disorder, the skyrmions form a pinned crystal and depin elastically, while for strong disorder the system forms a pinned amorphous state that depins plastically. At high drives the skyrmions can dynamically reorder into a moving crystal, with the onset of reordering determined by the strength of the Magnus term.