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.

We study experimentally and numerically a (quasi) two dimensional colloidal suspension of selfpropelled spherical particles. The particles are carbon-coated Janus particles, which are propelled due to diffusiophoresis in a near-critical water-lutidine mixture. At low densities, we find that the driving stabilizes small clusters. At higher densities, the suspension undergoes a phase separation into large clusters and a dilute gas phase. The same qualitative behavior is observed in simulations of a minimal model for repulsive self-propelled particles lacking any alignment interactions. The observed behavior is rationalized in terms of a dynamical instability due to the self-trapping of self-propelled particles.PACS numbers: 82.70. Dd,64.60.Cn Following our physical intuition, "agitating" a system by, e.g., increasing the temperature also increases disorder. The most simple and paradigmatic example is the Ising model of interacting spins on a lattice, which, in two or more dimensions, displays a second-order phase transition from an ordered state to a disordered state as we increase the temperature [1]. Non-equilibrium driven systems, however, may defy our intuition and show the opposite behavior: increasing the noise strength leads to the emergence of an ordered state [2,3], for example the "freezing by heating" transition of oppositely driven particles in a narrow channel [4].One class of non-equilibrium systems that currently receives considerable attention are self-propelled, or "active", particles [5][6][7][8][9][10][11][12][13]. These are model systems for "living active matter" ranging from microtubules [14] to dense bacterial solutions [15][16][17] to flocks of birds [18]. A common feature of many of these models is that the particle orientations align, which leads to a multitude of collective phenomena such as swarming [19] and even micro-bacterial turbulence [20]. This alignment interaction can be either explicit (Vicsek-type models [21]) or indirect. For example, in dense granular systems of rods [22] and disks [23], the combination of hardcore repulsion and propulsion implies an effective alignment. Somewhat surprisingly, recently it has been found that also self-propelled suspensions lacking any alignment mechanism are able to show collective behavior. Specifically, simulations of a minimal model for a suspension of repulsive disks below the freezing transition [24] show phase separation into a dense large cluster and a dilute gas phase [25,26]. Phase separation due to a densitydependent mobility has been discussed theoretically in the context of run-and-tumble bacteria [27], and a link has been made recently to self-propelled Brownian particles [28].Experimentally, active clustering of spherical colloidal particles has been observed for sedimenting, platinumcoated gold particles [10] and colloidal particles with an embedded hematite cube [13], where platinum and hematite act as catalysts for the decomposition of water peroxide. In both studies, aggregation is attributed to attractive forces. In thi...

Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.low Reynolds number swimming | velocity increment distributions | scaling S imple forms of life, like amoebae or bacteria, self-organize into remarkable macroscopic patterns (1, 2), ranging from extended networks (3, 4) to complex vortices (5-10) and swarms (11). These structures often bear a striking resemblance to assemblies of higher organisms [e.g., flocks of birds (12) or schools of fish (13,14)] and present important biological model systems to study nonequilibrium phases and their transitions (15)(16)(17). A particularly interesting manifestation of collective behavior in microbial suspensions is the emergence of meso-scale turbulent motion (7,8,18,19). Driven by the microorganisms' self-propulsion and their mutual interactions, such self-sustained "active turbulence" can have profound effects on nutrient mixing and molecular transport in microbiological systems (2,(20)(21)(22). However, in spite of recent progress (19,(23)(24)(25), the phenomenology of turbulent bacterial dynamics is scarcely understood and a commonly accepted theoretical description is lacking (2,16,26). The latter fact may not be surprising given that a comprehensive mathematical characterization of turbulence in conventional fluids has remained elusive after more than a century of intense research (27).In view of the various physical and chemical pathways through which bacteria may communicate (1, 11, 28), a basic yet unsolved problem is to identify those interactions...

Combining statistical-mechanical theories and neutron-scattering techniques, we show that the effective pair potential between star polymers is exponentially decaying for large distances and crosses over, at a density-dependent corona diameter, to an ultrasoft logarithmic repulsion for small distances. We also make the theoretical prediction that in concentrated star polymer solutions, this ultrasoft interaction induces an anomalous fluid structure factor which exhibits an unusually pronounced second peak.[S0031-9007(98)06148-1] PACS numbers: 61.25.Hq, 61.20.Gy, 82.70.Dd Star polymers consist of a well-defined number f of flexible polymer chains tethered to a central microscopic core. By enhancing this functionality (or arm number) f which governs the interpenetrability of two stars, one can continuously switch from unbranched polymer chains (f 1, 2) to a colloidal sphere (f ¿ 1). Hence, star polymers can actually be viewed as hybrids between polymerlike entities and colloidal particles establishing an important link between these different domains of physics. Moreover, star polymer solutions reveal quite a number of novel structural and dynamical properties which occur neither in single-chain polymers nor in suspensions of colloidal spheres; for recent reviews see Refs. [1,2].While the polymer conformations around a single star are well understood by computer simulation [3], scaling theory [4], and small-angle neutron scattering experiments [5], concentrated star polymer solutions are much more difficult to access due to the additional effective interactions between the stars. In particular, these interactions become relevant when the distance r between two star polymer centers is of the order of the so-called corona diameter s, which describes the spatial extension of the monomer density around a single star (see the inset of Fig. 1). This translates immediately into an overlap density r ‫ء‬ ϵ 1͞s 3 of the core number density r. Close to this overlap density r ‫ء‬ , there is an effective repulsion between stars resulting from the osmotic pressure arising between polymers from different cores. The repulsion is purely entropic; i.e., it simply scales with the thermal energy k B T . Witten and Pincus [6] were the first to derive the functional form of this repulsion. The effective potential between two stars, V ͑r͒, was found to depend logarithmically on r and to scale asymptotically as f 3͞2 with the arm number, i.e., V ͑r͒ 2k B T gf 3͞2 ln͑r͞s͒, where k B is Boltzmann's constant, T is the temperature, and g is an unknown numerical prefactor. Note that this result was obtained only for large f and for small distances r # s. Since this potential depends only weakly on r, the stars can be viewed as "ultrasoft" colloidal particles whose interaction is very different from common soft spheres described, e.g., by an inverse-power potential [7,8].The aim of this Letter is twofold: First, we describe the star polymer interaction quantitatively, proposing an explicit analytical expression for the effective pair potenti...