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THE RANDOM BATCH METHOD FOR N-BODY QUANTUM DYNAMICS

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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A Formal Passage From a System of Boltzmann Equations for Mixtures Towards a Vlasov-Euler System of Compressible Fluids

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Acta Mathematicae Applicatae Sinica 2019年第1期35卷 158-173页

作者： Laurent DESVILLETTES Fran?ois GOLSE Valeria RICCI Université Paris Diderot Sorbonne Paris Cité Institut de Mathématiques de Jussieu-Paris Rive GaucheUMR CNRS 7586 CNRS Sorbonne Universités UPMC Univ cmls Ecole polytechnique et CNRS Université Paris-Saclay Dipartimento di Matematica e Informatica Università degli Studi di Palermo

A formal asymptotics leading from a system of Boltzmann equations for mixtures towards either Vlasov-Navier-Stokes or Vlasov-Stokes equations of incompressible fluids was established by the same authors and Etienne Bernard in: A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory Commun. Math. Sci., 15: 1703–1741(2017) and A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures. KRM, 11: 43–69(2018). With the same starting point but with a different scaling, we establish here a formal asymptotics leading to the Vlasov-Euler system of compressible fluids. Explicit formulas for the coupling terms are obtained in two typical situations: for elastic hard spheres on one hand, and for collisions corresponding to the inelastic interaction with a macroscopic dust speck on the other hand.

关键词： Vlasov-Euler system Boltzmann equation Hydrodynamic limit Aerosols Sprays Gas mixture

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