Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics
File(s)
Accepted version
Author(s)
Degond, P
Frouvelle, A
Liu, J-G
Type
Journal Article
Abstract
We provide a complete and rigorous description of phase transitions for kinetic
models of self-propelled particles interacting through alignment. These models
exhibit a competition between alignment and noise. Both the alignment frequency
and noise intensity depend on a measure of the local alignment. We show that, in
the spatially homogeneous case, the phase transition features (number and nature of
equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded
in how the ratio between the alignment and noise intensities depend on the local
alignment. In the spatially inhomogeneous case, we derive the macroscopic models
associated to the stable equilibria and classify their hyperbolicity according to the
same function.
models of self-propelled particles interacting through alignment. These models
exhibit a competition between alignment and noise. Both the alignment frequency
and noise intensity depend on a measure of the local alignment. We show that, in
the spatially homogeneous case, the phase transition features (number and nature of
equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded
in how the ratio between the alignment and noise intensities depend on the local
alignment. In the spatially inhomogeneous case, we derive the macroscopic models
associated to the stable equilibria and classify their hyperbolicity according to the
same function.
Date Issued
2015-04-01
Date Acceptance
2014-09-19
Citation
Archive for Rational Mechanics and Analysis, 2015, 216 (1), pp.63-115
ISSN
0003-9527
Publisher
Springer
Start Page
63
End Page
115
Journal / Book Title
Archive for Rational Mechanics and Analysis
Volume
216
Issue
1
Copyright Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0800-7
Subjects
Science & Technology
Physical Sciences
Technology
Mathematics, Interdisciplinary Applications
Mechanics
Mathematics
MEAN-FIELD LIMIT
MACROSCOPIC LIMITS
FLOCKING DYNAMICS
DRIVEN PARTICLES
CONTINUUM-LIMIT
MODEL
SYSTEM
EQUATION
MOTION
Publication Status
Published
Date Publish Online
2014-10-07