Branching random walk solutions to the Wigner equation

07/03/2019
by   Sihong Shao, et al.
0

The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator Θ_V with an anti-symmetric kernel as the generator of two branches of jump processes, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition Θ_V=Θ^+_V-Θ^-_V, i.e., treating Θ_V as the difference of two positive operators Θ^±_V, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of Θ^±_V, instead of Θ_V. In other words, the decay of high-frequency components is totally ignored, resulting in a severe numerical sign problem. To fully utilize such decay property, we have recourse to the stationary phase approximation for Θ_V, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but gain a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/25/2020

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media (extended version)

In this article, we present new random walk methods to solve flow and tr...
research
04/07/2022

Symmetric cooperative motion in one dimension

We explore the relationship between recursive distributional equations a...
research
04/14/2021

A Gaussian fixed point random walk

In this note, we design a discrete random walk on the real line which ta...
research
08/21/2023

Characterization of random walks on space of unordered trees using efficient metric simulation

The simple random walk on ℤ^p shows two drastically different behaviours...
research
07/07/2022

Approximate Carathéodory bounds via Discrepancy Theory

The approximate Carathéodory problem in general form is as follows: Give...
research
05/15/2023

Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers

For certain materials science scenarios arising in rubber technology, on...
research
06/07/2022

The fundamental solution of a 1D evolution equation with a sign changing diffusion coefficient

In this work we investigate a 1D evolution equation involving a divergen...

Please sign up or login with your details

Forgot password? Click here to reset