A computable branching random walk for the many-body Wigner quantum dynamics

TL;DR

This paper introduces a stochastic branching random walk algorithm for simulating many-body Wigner quantum dynamics in phase space, providing a numerically efficient and accurate method validated by numerical experiments.

Contribution

It develops a novel probabilistic framework for the Wigner equation using branching random walks, enabling tractable quantum dynamics simulations.

Findings

The algorithm accurately simulates quantum scattering and systems.

It controls computational complexity via particle number growth rate estimation.

Numerical experiments confirm efficiency and validity.

Abstract

A branching random walk algorithm for the many-body Wigner equation and its numerical applications for quantum dynamics in phase space are proposed and analyzed. After introducing an auxiliary function, the (truncated) Wigner equation is cast into the integral formulation as well as its adjoint correspondence, both of which can be reformulated into the renewal-type equations and have transparent probabilistic interpretation. We prove that the first moment of a branching random walk happens to be the solution for the adjoint equation. More importantly, we detail that such stochastic model, associated with both importance sampling and resampling, paves the way for a numerically tractable scheme, within which the Wigner quantum dynamics is simulated in a time-marching manner and the complexity can be controlled with the help of an (exact) estimator of the growth rate of particle number. Typical numerical experiments on the Gaussian barrier scattering and a Helium-like system validate our theoretical findings, as well as demonstrate the accuracy, the efficiency and thus the computability of the Wigner branching random walk algorithm.