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Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics

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Title: Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics
Authors: Degond, P
Frouvelle, A
Liu, J-G
Item 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.
Issue Date: 1-Apr-2015
Date of Acceptance: 19-Sep-2014
URI: http://hdl.handle.net/10044/1/26832
DOI: 10.1007/s00205-014-0800-7
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
Keywords: Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
MEAN-FIELD LIMIT
MACROSCOPIC LIMITS
FLOCKING DYNAMICS
DRIVEN PARTICLES
CONTINUUM-LIMIT
MODEL
SYSTEM
EQUATION
MOTION
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
math.AP
math.AP
math-ph
math.MP
0101 Pure Mathematics
0102 Applied Mathematics
General Physics
Publication Status: Published
Online Publication Date: 2014-10-07
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences