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−...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.展开更多
基金The work of Shi Jin was partly supported by NSFC grants No.11871297 and No.31571071We thank E.Moulines for kindly indicating several references on stochastic approximation.
文摘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.
