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On the mean field limit of the random batch method for interacting particle systems

Science China Mathematics 2022年第1期65卷 169-202页

作者： Shi Jin Lei Li School of Mathematical Sciences Institute of Natural SciencesMOE-LSCShanghai Jiao Tong UniversityShanghai 200240China

The random batch method proposed in our previous work(Jin et al.J Comput Phys,2020)is not only a numerical method for interacting particle systems and its mean-field limit,but also can be viewed as a model of the particle system in which particles interact,at discrete time,with randomly selected mini-batch of *** this paper,we investigate the mean-field limit of this model as the number of particles N→∞.Unlike the classical mean field limit for interacting particle systems where the law of large numbers plays the role and the chaos is propagated to later times,the mean field limit now does not rely on the law of large numbers and the chaos is imposed at every discrete *** this,we will not only justify this mean-field limit(discrete in time)but will also show that the limit,as the discrete time intervalτ→0,approaches to the solution of a nonlinear Fokker-Planck equation arising as the mean-field limit of the original interacting particle system in the Wasserstein distance.

关键词： random batch method mean field limit chaos Wasserstein distance nonlinear Fokker-Planck equation

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THE random batch method FOR N-BODY QUANTUM DYNAMICS

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Journal of Computational Mathematics 2021年第6期39卷 897-922页

作者： Francois Golse Shi Jin Thierry Paul Ecole polytechnique CMLS91128 Palaiseau CedexFrance School of Mathematical Sciences Institute of Natural SciencesMOE-LSCShanghai Jiao Tong UniversityShanghai 200240China Sorbonne Universite CNRSUniversite de ParisINRIALaboratoire Jacques-Louis Lions75005 ParisFrance

This paper discusses a numerical method for computing the evolution of large interacting system of quantum *** 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)/*** 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 *** 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

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