Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs
Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs
复制标题
DOI：
10.1007/s00205-019-01436-y
复制
发表时间：
2018-10
影响因子：
2.5
通讯作者：
Adriana Garroni;Patrick van Meurs;M. Peletier;Lucia Scardia
中科院分区：
文献类型：
--
作者：
Adriana Garroni;Patrick van Meurs;M. Peletier;Lucia Scardia
我们解决的问题，不断发展的相互作用的粒子系统的粒子数趋于无穷大的收敛。我们考虑两种类型的粒子，称为正粒子和负粒子。同号粒子互相排斥，异号粒子互相吸引。相互作用势对所有粒子都是相同的，直到符号为止，并且在零处具有对数奇点。这种系统的主要例子是晶体中的位错。由于相互作用势的奇异性，离散演化导致有限时间内的爆破。我们通过在长度尺度上正则化相互作用势来补救这种情况，当粒子数趋于无穷大时，相互作用势收敛到零。我们建立两个主要结果。第一个是一个进化收敛的结果表明，经验措施的积极和消极的粒子收敛到一组耦合偏微分方程的解决方案，描述其连续密度的演变。在位错的情况下，这些偏微分方程被称为Groma-Balogh方程。在证明中，我们依赖于凸梯度流理论，建立一个定量的约束之间的距离的经验措施和连续的解决方案，正规化版本的Groma-Balogh方程，prioranda估计的Groma-Balogh方程传递到小正规化限制在一个功能设置的基础上Orlicz空间。为了使定量界不退化太快的限制，我们需要收敛到零足够缓慢。第二个结果是一个反例，证明了如果ε快速收敛到零，则正位错和负位错的经验测度的极限不满足Groma-Balogh方程。这些结果表明，格罗马-巴洛格方程作为多粒子系统极限的有效性如何以微妙的方式取决于势的奇异性被正则化的尺度。
We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale, which converges to zero as the number of particlesntends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma–Balogh equations. In the proof we rely on both the theory of-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a-regularised version of the Groma–Balogh equations, anda prioriestimates for the Groma–Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limitwe requireto converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that ifconverges to zero sufficientlyfast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma–Balogh equations. These results show how the validity of the Groma–Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.