The Random Batch Method for $N$-Body Quantum Dynamics

Abstract

This paper discusses a numerical method for computing the evolution of large interacting system of quantum particles. The 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 $p\ll N$ particles chosen at random at each time step, multiplied by $(N-1)/p$. This 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 $p=1$ \u2014 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 step. We 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 $\hbar$. 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 $\mathbf{R}^d$ used in the context of optimal transport.