Speaker: Professor Shao Sihong, Peking University

Date: 2016/12/15 Thursday 10:30-11:30AM 

Venue: 数学楼二楼学术报告厅 

Abstract: As a phase space language for quantum mechanics, the Wigner function approaches bear a close analogy to classical mechanics and have been drawing growing attention, especially in simulating quantum many-body systems. In this talk, we will summarize recent progress in numerical methods for the time dependent Wigner equation. 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.