# A Branching Random Walk Method for Many-Body Wigner Quantum Dynamics

@article{Shao2019ABR, title={A Branching Random Walk Method for Many-Body Wigner Quantum Dynamics}, author={Sihong Shao and Yunfeng Xiong}, journal={Numerical Mathematics: Theory, Methods and Applications}, year={2019} }

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…

## 10 Citations

### The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation

- Computer ScienceCommunications in Computational Physics
- 2019

An improvement of the first signed-particle implementation that partially alleviates the restriction on the time step is provided and a thorough theoretical and numerical comparison among all the existing signed- particle implementations is performed.

### Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation

- Computer Science
- 2020

An adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem and attempt to simulate the proton-electron couplings in 6-D and 12-D phase space.

### Overcoming the numerical sign problem in Wigner dynamics via particle annihilation

- Computer Science
- 2020

This paper makes the first attempt to simulate the transitions of hydrogen energy levels in 6-D phase space, where the feasibility of PAUM with sample sizes about $10^9$-$10^{10}$ has also been explored as a comparison.

### Branching Random Walk Solutions to the Wigner Equation

- MathematicsSIAM J. Numer. Anal.
- 2020

The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $\Theta_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed and it is proved that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $\The ta_V$.

### A computational approach for investigating Coulomb interaction using Wigner–Poisson coupling

- PhysicsJournal of computational electronics
- 2021

This work reduces the computational complexity of the time evolution of two interacting electrons by resorting to reasonable approximations and demonstrates that the entanglement due to the Coulomb interaction is well accounted for by the introduced local approximation.

### A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics

- Physics
- 2022

. Numerical resolution for 6-D Wigner dynamics under the Coulomb potential faces with the combined challenges of high dimensionality, nonlocality, oscillation and singularity. In par-ticular, the…

### Efficient implementation and performance evaluation of the Wigner branching random walk

- Computer Science
- 2017

Two efficient strategies to realize the signed-particle implementation are proposed, one to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other to utilize a bootstrap filter to adjust the skewness of particle weights.

### Solving the Wigner equation with signed particle Monte Carlo for chemically relevant potentials.

- PhysicsThe Journal of chemical physics
- 2021

This paper adapts the signed particles Monte Carlo algorithm for solving the transient Wigner equation to scenarios of chemical interest and demonstrates its excellent performance on harmonic and double well potentials for electronic and nuclear systems.

### SPADE: Sequential-clustering Particle Annihilation via Discrepancy Estimation

- Computer Science, MathematicsArXiv
- 2020

An algorithm for PA in high-dimensional Euclidean space based on hybrid of clustering and matching, dubbed the Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), implying that SPADE can be immune to the curse of dimensionality for a wide class of test functions.

## References

SHOWING 1-10 OF 53 REFERENCES

### A computable branching process for the Wigner quantum dynamics

- Physics, Mathematics
- 2016

A branching process treatment for the nonlocal Wigner pseudo-differential operator and its numerical applications in quantum dynamics is proposed and analyzed. We start from the discussion on two…

### Solution of the Space-dependent Wigner Equation Using a Particle Model

- PhysicsMonte Carlo Methods Appl.
- 2004

In this work, the stationary, position-dependent Wigner equation is considered and particle models are presented which interpret the potential operator as a generation term of numerical particles of positive and negative statistical weight.

### The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation

- Computer ScienceCommunications in Computational Physics
- 2019

An improvement of the first signed-particle implementation that partially alleviates the restriction on the time step is provided and a thorough theoretical and numerical comparison among all the existing signed- particle implementations is performed.

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

- PhysicsSIAM J. Sci. Comput.
- 2016

This paper serves as the first attempt to solve the time-dependent many-body Wigner equation through a grid-based advective-spectral-mixed method and can also rigorously preserve physical symmetry relations.

### The Wigner representation of quantum mechanics

- Physics
- 1983

Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and…

### Comparison of deterministic and stochastic methods for time-dependent Wigner simulations

- PhysicsJ. Comput. Phys.
- 2015

### Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices

- Physics
- 2004

Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The…

### A benchmark study of the Wigner Monte Carlo method

- PhysicsMonte Carlo Methods Appl.
- 2014

This paper extends the second approach to the Wigner equation for time-dependent simulations and presents a validation against a well-known benchmark model, the Schrödinger equation, demonstrating excellent quantitative agreement.

### DEFORMATION QUANTIZATION: QUANTUM MECHANICS LIVES AND WORKS IN PHASE-SPACE

- PhysicsInternational Journal of Modern Physics A
- 2002

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear…