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A NON-LINEAR KINETIC MODEL OF SELF-PROPELLED PARTICLES WITH MULTIPLE EQUILIBRIA

机译：具有多重均衡的自推进粒子的非线性动力学模型

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摘要

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density f(t), in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only hypoelliptic) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate N-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as N -> infinity, to the solution of the PDE.
机译：我们介绍和分析维基克型互动粒子系统的连续模型。该模型由非线性动力学部分微分方程（PDE）给出，所述非线性动力学部分微分方程（PDE）描述了密度F（t），在单个粒子相空间中的密度f（t）的时间演变，所述间隔空间的相互作用颗粒的集合中被限制在所述尺寸圆环。粒子的位置和速度的相应随机微分方程是条件Mckean-Vlasov类型的演化（条件是该过程通过自己的条件期望取决于其自身法律）。在本文中，我们考虑了PDE解决方案的存在性和唯一性。从PDE既不椭圆（线性部分仅是低管）也不是梯度形式的挑战。此外，对于相互作用功能的一些特定选择和密度分布不​​依赖于空间变量的简化情况，我们表明该模型呈现多个固定状态（对应于形成协调顺时针/逆时针/逆时针旋转运​​动的粒子）我们也研究了这些国家的融合。最后，我们证明了适当的N-粒子系统的平均局部收敛到我们的PDE的解决方案：更确切地说，我们表明这种粒子系统的经验测量弱，如n - >无限远，到了PDE。

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