Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Frouvelle, Amic and Liu, Jian-Guo (2012) Dynamics in a Kinetic Model of Oriented Particles with Phase Transition. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 4 (2). pp. 791-826. ISSN 0036-1410

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Official URL: http://epubs.siam.org/doi/abs/10.1137/110823912

Abstract

Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing more than the Smoluchowski equation on the sphere with dipolar potential. In this self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, the other equilibria are the Fisher–von Mises distributions with arbitrary direction and a concentration parameter determined by the intensity of the noise. For any initial condition, we give a rigorous proof of convergence of the solution to a steady state as time goes to infinity. In particular, when the noise is under the threshold and with nonzero initial mean velocity, the solution converges exponentially fast to a unique Fisher–von Mises distribution. We also found a new conservation relation, which can be viewed as a convex quadratic entropy when the noise is above the threshold. This provides a uniform exponential rate of convergence to the uniform distribution. At the threshold, we show algebraic decay to the uniform distribution.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Engineering, Science and Mathematics > School of Mathematics > Department of Applied Mathematics
Depositing User: Amic Frouvelle
Date Deposited: 06 Oct 2012 09:08
Last Modified: 07 May 2017 19:33
URI: http://preprints.acmac.uoc.gr/id/eprint/147

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