# Random Batch Methods for classical and quantum N-body problems

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.