Random Batch Methods for classical and quantum N-body problems

Abstract

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 $\mathcal{O}(N^2)$ per time step to $\mathcal{O}(N)$, 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. This method is also extended to quantum Monte - Carlo methods for N - body Schrodinger equation and will be shown to have significant gains in computational speed up over the classical Metropolis - Hastings algorithm and the Langevin dynamics based Euler - Maruyama method for statistical samplings of general distributions for interacting particles. For quantum N - body Schrodinger equation, we also 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). This talk is based on joint works with Lei Li, Jian - Guo Liu, Francois Golse, Thierry Paul and Xiantao Li.