We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions.
For quantum N-body Schrodinger equation, we obtain, for pair-wise random interactions, a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is uniform in N > 1 and independent of the Planck constant \hbar. To this goal we need to introduce a new metric specially tailored to handle at the same time the difficulties pertaining to the small \hbar regime (classical limit), and those pertaining to the large N regime (mean-field limit).
The classical part was a joint work with Lei Li and Jian-Guo Liu, while the quantum part was with Francois Golse and Thierry Paul.
This is a joint Math Department-ICM Colloquium lecture.
Jin Shi is the Director of Institute of Natural Sciences, and Chair Professor of Mathematics, at Shanghai Jiao Tong University. He obtained his BS degree from Peking University and his Ph.D. from University of Arizona. He was a postdoc at Courant Institute, New York University, an assistant and associate professors at Georgia Institute of Technology, and full professor, department chair and Vilas Distinguished Achievement Professor at University of Wisconsin-Madison, Chair of Department of Mathematics at Shanghai Jiao Tong University.
He also serves as a co-director of the Shanghai Center of Applied Mathematics, director of Ministry of Education Key Lab on Scientific and Engineering Computing, and director of Center for Mathematical Foundation of Artificial Intelligence at Shanghai Jiao Tong University.
He received a Feng Kang Prize of Scientific Computing in 2001., He is an inaugural Fellow of the American Mathematical Society (AMS) (2012), a Fellow of Society of Industrial and Applied Mathematics (SIAM) (2013), and an Invited Speaker at the International Congress of Mathematicians in 2018.
His research interests include kinetic theory, quantum dynamics, uncertainty quantification, interacting particle systems and computational fluid dynamics, etc. He has published over 170 research papers, and was awarded one of the four Best Paper Awards by the Springer Journal Research in the Mathematical Sciences for its fifth year anniversary.