Dynamics in a Kinetic Model of Oriented Particles with Phase Transition
Dynamics in a Kinetic Model of Oriented Particles with Phase Transition
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DOI：
10.1137/110823912
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发表时间：
2011-01
期刊：
影响因子：
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通讯作者：
A. Frouvelle;Jian‐Guo Liu
中科院分区：
文献类型：
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作者：
A. Frouvelle;Jian‐Guo Liu
受自推进粒子排列模型中的相变现象的启发，我们得到了一个动力学平均场方程，它就是带有偶极势的Doi方程（也称为Smoluchowski方程）。在一篇自成一体的文章中，我们只使用基本的工具，分析了这个方程在任何维度上的动态。我们首先证明了这个方程的整体适定性，在任何Sobolev空间的初始条件。然后我们计算所有可能的稳态。噪声参数有一个阈值：超过这个阈值，唯一的平衡是均匀分布，在这个阈值之下，还有一个非各向同性平衡族。我们给出了严格的证明，收敛到一个稳定的状态，随着时间的推移到无穷大的解决方案。特别是，我们表明，在超临界的情况下，唯一的初始条件，导致在大的时间均匀分布是那些消失的动量。对于任何正值的噪声参数，和任何初始条件下，我们给出的收敛速度向平衡，指数为超临界和亚临界的情况下，代数的临界情况下。
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 else than the Doi equation (also called Smoluchowski equation) with dipolar potential. In a 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, there is also a family of non-isotropic equilibria. We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.