Yunfeng Xiong

Yunfeng Xiong, Zhenzhu Chen, Sihong Shao

As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic numerical solutions have been almost exclusively confined to one-dimensional one-body systems and few results are reported even for one-dimensional two-body problems. This paper serves as the first attempt to solve the time-dependent many-body Wigner equation through a grid-based advective-spectral-mixed method. The main feature of the method is to resolve the linear advection in $(\bm{x},t)$-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in $\bm k$-space is adopted for accurate calculation of the nonlocal pseudo-differential term. Not only the time step of the resulting method is not restricted by the usual CFL condition and thus a large time step is allowed, but also the mass conservation can be maintained. In particular, for the system consisting of identical particles, the advective-spectral-mixed method can also rigorously preserve physical symmetry relations. The performance is validated through several typical numerical experiments, like the Gaussian barrier scattering, electron-electron interaction and a Helium-like system, where the third-order accuracy against both grid spacing and time stepping is observed.

Zhenzhu Chen, Yunfeng Xiong, Sihong Shao

Unbounded potentials are always utilized to strictly confine quantum dynamics and generate bound or stationary states due to the existence of quantum tunneling. However, the existed accurate Wigner solvers are often designed for either localized potentials or those of the polynomial type. This paper attempts to solve the time-dependent Wigner equation in the presence of a general class of unbounded potentials by exploiting two equivalent forms of the pseudo-differential operator: integral form and series form (i.e., the Moyal expansion). The unbounded parts at infinities are approximated or modeled by polynomials and then a remaining localized potential dominates the central area. The fact that the Moyal expansion reduces to a finite series for polynomial potentials is fully utilized. Using a spectral collocation discretization which conserves both mass and energy, several typical quantum systems are simulated with a high accuracy and reliable estimation of macroscopically measurable quantities is thus obtained.

Sihong Shao, Yunfeng Xiong

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.

Rui Wang, Yunfeng Xiong, Zhengru Zhang

The chemotaxis PDE system with singular sensitivity was originally proposed by Short et al. (Math. Mod. Meth. Appl. Sci., 2008) as the continuum limit of a biased random walk model to account for the formation of crime hotspots and environmental feedback successfully. Recently, this idea has also been applied to epidemiology to model the impact of human social behaviors on disease transmission. In order to characterize the phase transition, pattern formation and statistical properties in the long-term dynamics, a stable and accurate numerical scheme is urgently demanded, which still remains challenging due to the positivity constraint on the singular sensitivity and the absence of an energy functional. In particular, the loss of positivity may produce nonphysical states and even cause spurious blow-up. To address these numerical challenges, this paper constructs an efficient positivity-preserving, implicit-explicit scheme with second-order accuracy. A rigorous error estimation is provided with the Lagrange multiplier correction to deal with the singular sensitivity. The whole framework is extended to a multi-agent epidemic model with degenerate diffusion, in which both positivity and mass conservation are achieved. Numerical experiments are performed to validate the theoretical results and demonstrate the necessity of the correction strategy. Our simulations reveal rich dynamical behaviors, including the phase transition between aggregation-dominated and dissipative regimes, as well as the nucleation, spread, and dissipation of crime hotspots. For the epidemic model, the results further show that spatial clustering of population density may accelerate virus transmission and significantly amplify the infectious wave.