An Advective-Spectral-Mixed Method for Time-Dependent Many-Body Wigner Simulations

Abstract

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 (x, t)-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in 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.

Numerical Methods for the Wigner Equation with Unbounded Potential

“…Here we assume that V w decays and thus ignore the periodic images. A necessary and sufficient condition for the truncated Wigner equation (3.1) to conserve the mass has been given in [13] and reads L k ∆y = 2π, (3.4) where L k = k max − k min represents the length of k-domain. In x-space, the popular quantum transitive boundary condition will be adopted hereafter as did in [12].…”

“…where n(x, t) is the particle density and j(x, t) the current density [13]. The other is the energy conservation…”

Efficient solution of the Wigner–Liouville equation using a spectral decomposition of the force field

“…Pseudo-spectral methods have been successfully used to accurately treat the Wigner function in Refs. [16,17]. However, in this section, we do not present such a highly optimized implementation.…”

“…While having been studied for a long time [14], direct discretization using well-designed finite difference approaches has recently been improved by using the weighted essentially non-oscillating scheme (WENO) to prevent erroneous oscillations [15]. In other work, it has also been shown that the Wigner function can be solved in a robust way using adaptive pseudo-spectral methods in the form of massconserving spectral element methods [16,17]. In these methods, the carefully crafted spectral decomposition of the Wigner function enables the oscillatory components introduced by the Wigner kernel to be solved exactly.…”

“…In this paper, we propose a deterministic method to solve the Wigner equation by exploiting the analogy with the classical Boltzmann transport equation without, however, invoking the classical limit. In contrast to the highly optimized numerical techniques that have been outlined above [6,[10][11][12][13][15][16][17], our main focus is on the development of a different, but equivalent form of the Wigner function that allows for a natural, direct implementation. To this end, we recast the Wigner equation into a form that employs the force field rather than the potential energy from which one would conventionally proceed in quantum mechanics.…”