Discrete-to-Continuum Convergence of Charged Particles in 1D with Annihilation
Discrete-to-Continuum Convergence of Charged Particles in 1D with Annihilation
复制标题
一维带电粒子湮灭的离散到连续收敛
DOI：
10.1007/s00205-022-01812-1
复制
发表时间：
2022
影响因子：
2.5
通讯作者：
Pozar Norbert
中科院分区：
文献类型：
--
作者：
van Meurs Patrick;Peletier Mark A.;Pozar Norbert
我们考虑一个由静电相互作用驱动的带电粒子在真实的直线上运动的系统。由于我们考虑两种符号的电荷，碰撞可能在有限时间内发生。在碰撞时，一些碰撞粒子被有效地从系统中移除（湮灭）。我们想到的两个应用是金属中的涡旋和位错。在本文中，我们实现了两个目标。首先，我们发展了具有湮灭的相互作用粒子系统的严格解的概念。这里的主要创新是提供了一个仔细的管理组的两个以上的粒子的湮灭，我们证明了定义是一致的，通过证明存在性，唯一性和连续依赖于初始数据。证明依赖于一个详细的分析ODE接近碰撞的轨迹，并reparametrization的矢量的时刻，他们的元素。其次，我们传递到多粒子极限（离散到连续），并恢复预期的粒子密度的极限方程。由于奇异的相互作用和湮灭规则，离散到连续极限的标准证明技术不适用。特别是，措施框架似乎不合适。相反，我们使用的一维功能，粒子系统和限制PDE可以在哈密尔顿-雅可比方程的特点。虽然我们的证明遵循一个标准的限制程序，这样的方程，相对于现有的结果的新奇在于允许更强的奇异性的粒子系统中，利用自由选择的定义中的粘度解决方案。
We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are effectively removed from the system (annihilation). The two applications we have in mind are vortices and dislocations in metals. In this paper we achieve two goals. First, we develop a rigorous solution concept for the interacting particle system with annihilation. The main innovation here is to provide a careful management of the annihilation of groups of more than two particles, and we show that the definition is consistent by proving existence, uniqueness, and continuous dependence on initial data. The proof relies on a detailed analysis of ODE trajectories close to collision, and a reparametrization of vectors in terms of the moments of their elements. Second, we pass to the many-particle limit (discrete-to-continuum), and recover the expected limiting equation for the particle density. Due to the singular interactions and the annihilation rule, standard proof techniques of discrete-to-continuum limits do not apply. In particular, the framework of measures seems unfit. Instead, we use the one-dimensional feature that both the particle system and the limiting PDE can be characterized in terms of Hamilton–Jacobi equations. While our proof follows a standard limit procedure for such equations, the novelty with respect to existing results lies in allowing for stronger singularities in the particle system by exploiting the freedom of choice in the definition of viscosity solutions.