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

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