Motivated by aggregation phenomena in gliding bacteria, we study collective motion in a twodimensional model of active, self-propelled rods interacting through volume exclusion. In simulations with individual particles, we find that particle clustering is facilitated by a sufficiently large packing fraction η or length-to-width ratio κ. The transition to clustering in simulations is well captured by a mean-field model for the cluster size distribution, which predicts that the transition values κc of the aspect ratio for a fixed packing fraction η is given by κc = C/η − 1 where C is a constant. Introduction.-Emergent large-scale patterns of interacting self-driven motile elements are observed in a wide range of biological systems of different complexity: from human crowds, herds, bird flocks, and fish schools [1] to multicellular aggregates, e.g. of bacteria and amoebae [2] as well as sperms [3]. A recurrent question is how these entities coordinate their behavior to form groups which move collectively. At a theoretical level, several qualitative approaches have been made to incorporate the diverse collective behaviors of such different systems in a common framework [1,4,5]. More specific models for bacteria like E. coli as well as for amoebae like D. discoideum [2], have been based on chemotaxis, a long-range cell interaction mechanism according to which individual cells move in response to chemical signals produced by all other cells. However, in some bacteria there is no evidence for chemotactic cues and cells coordinate their movement by cell-to-cell signalling mechanisms in which physical contact between bacteria is needed [6]. Consequently, one may ask how such bacteria aggregate in order to communicate.

We study, in two space dimensions, the large-scale properties of collections of constant-speed polar point particles interacting locally by nematic alignment in the presence of noise. This minimal approach to self-propelled rods allows one to deal with large numbers of particles, revealing a phenomenology previously unseen in more complicated models, and moreover distinctively different from both that of the purely polar case (e.g. the Vicsek model) and of active nematics.PACS numbers: 05.65.+b, 87.18.Hf, 87.18.Gh Collective motion is an ubiquitous phenomenon observable at all scales, in natural systems [1] as well as human societies [2]. The mechanisms at its origin can be remarkably varied. For instance, they may involve the hydrodynamic interactions mediated by the fluid in which bacteria swim [3], the long-range chemical signaling driving the formation and organization of aggregation centers of Dictyostelium discoideum amoeba cells [4], or the local cannibalistic interactions between marching locusts [5]. In spite of this diversity, one may search for possible universal features of collective motion, a context in which the study of "minimal" models is a crucial step. Recently, the investigation of the simplest cases, where the problem is reduced to the competition between a local aligning interaction and some noise, has revealed a wealth of unexpected collective properties. For example, constant speed, self-propelled, polar point particles with ferromagnetic interactions subjected to noise (as in the Vicsek model [6]) can form a collectively moving fluctuating phase with long-range polar order even in two spatial dimensions [7], with striking properties such as spontaneous segregation into ordered solitary bands moving in a sparse, disordered sea, or anomalous ("giant") density fluctuations [8]. In contrast, active apolar particles with nematic interactions only exhibit quasi-long-range nematic order in two dimensions with segregation taking the form of a single, strongly-fluctuating, dense structure with longitudinal order and even stronger density fluctuations than in the polar-ferromagnetic case [9,10,11].Noting that these differences reflect those in the local symmetry of particles and their interactions, a third situation can be defined, intermediate between the polar ferromagnetic model and the apolar nematic one, that of self-propelled polar particles aligning nematically [12]. Such a mechanism is typically induced by volume exclusion interactions, when elongated particles colliding almost head-on slide past each other, as illustrated schematically in Fig. 1. Thus, self-propelled polar point particles with apolar interactions can be conceived as a minimal model for self-propelled rods interacting by inelastic collisions [13,14,15]. Other relevant situations can be found in a biological context, such as gliding myxobacteria moving on a substrate [16], or microtubules driven by molecular motors grafted on a surface [17].In this Letter, we study collections of constant-speed polar point particles inte...

We characterize cell motion in experiments and show that the transition to collective motion in colonies of gliding bacterial cells confined to a monolayer appears through the organization of cells into larger moving clusters. Collective motion by nonequilibrium cluster formation is detected for a critical cell packing fraction around 17%. This transition is characterized by a scale-free power-law cluster-size distribution, with an exponent 0.88±0.07, and the appearance of giant number fluctuations. Our findings are in quantitative agreement with simulations of self-propelled rods. This suggests that the interplay of self-propulsion and the rod shape of bacteria is sufficient to induce collective motion.

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Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of "active matter" in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano-and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics.

Coherent vortical motion has been reported in a wide variety of populations including living organisms (bacteria, fishes, human crowds) and synthetic active matter (shaken grains, mixtures of biopolymers), yet a unified description of the formation and structure of this pattern remains lacking. Here we report the self-organization of motile colloids into a macroscopic steadily rotating vortex. Combining physical experiments and numerical simulations, we elucidate this collective behaviour. We demonstrate that the emergent-vortex structure lives on the verge of a phase separation, and single out the very constituents responsible for this state of polar active matter. Building on this observation, we establish a continuum theory and lay out a strong foundation for the description of vortical collective motion in a broad class of motile populations constrained by geometrical boundaries.

Abstract. A mean-field approach (MFA) is proposed for the analysis of orientational order in a two-dimensional system of stochastic self-propelled particles interacting by local velocity alignment mechanism. The treatment is applied to the cases of ferromagnetic (F) and liquid-crystal (LC) alignment. In both cases, MFA yields a second order phase transition for a critical noise strength and a scaling exponent of 1/2 for the respective order parameters. We find that the critical noise amplitude ηc at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. η LC C < η F C . A comparison with simulations of individual-based models with F-resp. LC-alignment shows that the predictions about the critical behavior and the qualitative relation between the respective critical noise amplitudes are correct.

We study general aspects of active motion with fluctuations in the speed and the direction of motion in two dimensions. We consider the case in which fluctuations in the speed are not correlated to fluctuations in the direction of motion, and assume that both processes can be described by independent characteristic time scales. We show the occurrence of a complex transient that can exhibit a series of alternating regimes of motion, for two different angular dynamics which correspond to persistent and directed random walks. We also show additive corrections to the diffusion coefficient. The characteristic time scales are also exposed in the velocity autocorrelation, which is a sum of exponential forms.