THE RANDOM BATCH METHOD FOR N-BODY QUANTUM DYNAMICS

• 作者: Fran(c)ois Golse Shi Jin Thierry Paul
• 来源期刊: 计算数学（英文版）
• 年，卷（期）: 2021,（6）
• 页码范围: 897-922
• 页数: 26页
• 语种: 英文
• DOI: 10.4208/jcm.2107-m2020-0306

摘要

This paper discusses a numerical method for computing the evolution of large inter-acting system of quantum particles.The idea of the random batch method is to replace the total interaction of each particle with the N-1 other particles by the interaction with p ＜＜ N particles chosen at random at each time step,multiplied by (N-1)/p.This re-duces the computational cost of computing the interaction potential per time step from O(N2) to O(N).For simplicity,we consider only in this work the case p =1 — in other words,we assume that N is even,and that at each time step,the N particles are orga-nized in N/2 pairs,with a random reshuffling of the pairs at the beginning of each time step.We obtain a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is both uniform in N ＞ 1 and independent of the Planck constant h.The key idea is to use a new type of distance on the set of quantum states that is reminiscent of the Wasserstein distance of exponent 1 (or Monge-Kantorovich-Rubinstein distance) on the set of Borel probability measures on Rd used in the context of optimal transport.