Mean-field theory of collective motion due to velocity alignment

We introduce a system of self-propelled agents (active Brownian particles) with velocity alignment in two spatial dimensions and derive a mean-field theory from the microscopic dynamics via a nonlinear Fokker-Planck equation and a moment expansion of the probability distribution function. We analyze the stationary solutions corresponding to macroscopic collective motion with finite center of mass velocity (ordered state) and the disordered solution with no collective motion in the spatially homogeneous system. In particular, we discuss the impact of two different propulsion functions governing the individual dynamics. Our results predict a strong impact of the individual dynamics on the mean field onset of collective motion (continuous vs discontinuous). In addition to the macroscopic density and velocity field we consider explicitly the dynamics of an effective temperature of the agent system, representing a measure of velocity fluctuations around the mean velocity. We show that the temperature decreases strongly with increasing level of collective motion despite constant fluctuations on individual level, which suggests that extreme caution should be taken in deducing individual behavior, such as, state-dependent individual fluctuations from mean-field measurements [Yates {\em et al.}, PNAS, 106 (14), 2009].Comment: corrected version, Ecological Complexity (2011) in pres

Swarming and Pattern Formation due to Selective Attraction and Repulsion

We discuss the collective dynamics of self-propelled particles with selective attraction and repulsion interactions. Each particle, or individual, may respond differently to its neighbors depending on the sign of their relative velocity. Thus, it is able to distinguish approaching (coming closer) and moving away individuals. This differentiation of the social response is motivated by the response to looming visual stimuli and may be seen as a generalization of the previously proposed, biologically motivated, escape and pursuit interactions. The model can account for different types of behavior such as pure attraction, pure repulsion, or escape and pursuit depending on the values (signs) of the different response strengths, and provides, in the light of recent experimental results, an interesting alternative to previously proposed models of collective motion with an explicit velocity-alignment interaction. We show the onset of large scale collective motion in a subregion of the parameter space, which corresponds to an effective escape and/or pursuit response. Furthermore, we discuss the observed spatial patterns and show how kinetic description of the dynamics can be derived from the individual based model.Comment: Preprint, 24 pages, submitted to Interface Focu

Self-organized escape processes of linear chains in nonlinear potentials

An enhancement of localized nonlinear modes in coupled systems gives rise to a novel type of escape process. We study a spatially one dimensional set-up consisting of a linearly coupled oscillator chain of $N$ mass-points situated in a metastable nonlinear potential. The Hamilton-dynamics exhibits breather solutions as a result of modulational instability of the phonon states. These breathers localize energy by freezing other parts of the chain. Eventually this localised part of the chain grows in amplitude until it overcomes the critical elongation characterized by the transition state. Doing so, the breathers ignite an escape by pulling the remaining chain over the barrier. Even if the formation of singular breathers is insufficient for an escape, coalescence of moving breathers can result in the required concentration of energy. Compared to a chain system with linear damping and thermal fluctuations the breathers help the chain to overcome the barriers faster in the case of low damping. With larger damping, the decreasing life time of the breathers effectively inhibits the escape process.Comment: 14 pages, 13 figure

Self-propelled particles with selective attraction-repulsion interaction - From microscopic dynamics to coarse-grained theories

In this work we derive and analyze coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of a one-particle density function. This approach allows effective numerical investigation of the stability of possible solutions of the system. The detailed analysis of the interaction integrals entering the equations demonstrates that divergences at small wavelengths can appear at arbitrary expansion orders. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. Finally, we compare our analytical predictions on the stability of the disordered homogeneous solution with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. Disagreements of numerical results and the hydrodynamic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present Toner and Tu type theory of polar active matter.Comment: revised version, 37pages, 11 figure