solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看5809次
Convergence and non-convergence of many-particle evolutions with multiple signs. (arXiv:1810.04934v1 [math.AP])
来源于:arXiv
We address the question of 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 $\delta_n>0$, which
converges to zero as the number of particles $n$ tends 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 查看全文>>