312 research outputs found
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 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
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