THE RANDOM BATCH METHOD FOR N-BODY QUANTUM DYNAMICS  

作　　者：Francois Golse Shi Jin Thierry Paul 

机构地区：[1]Ecole polytechnique,CMLS,91128 Palaiseau Cedex,France [2]School of Mathematical Sciences,Institute of Natural Sciences,MOE-LSC,Shanghai Jiao Tong University,Shanghai 200240,China [3]Sorbonne Universite,CNRS,Universite de Paris,INRIA,Laboratoire Jacques-Louis Lions,75005 Paris,France

出　　处：《Journal of Computational Mathematics》2021年第6期897-922,共26页计算数学（英文）

基　　金：The work of Shi Jin was partly supported by NSFC grants No.11871297 and No.31571071;We thank E.Moulines for kindly indicating several references on stochastic approximation.

摘　　要：This paper discusses a numerical method for computing the evolution of large interacting 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 reduces the computational cost of computing the interaction potential per time step from O(N^(2))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 organized 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 h̵.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.

关 键 词：Time-dependent Schrodinger equations Random batch method Mean-field limit Wasserstein distance 

分 类 号：O17[理学—数学]