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

Yunfeng Xiong, Sihong Shao

Abstract

The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D $\le$ 4, the mesh size grows dramatically when D $\ge$ 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a $73^3 \times 80^3$ uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers.

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

Abstract

The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D 4, the mesh size grows dramatically when D 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers.

Paper Structure

This paper contains 44 sections, 2 theorems, 100 equations, 38 figures, 11 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum two-body Coulomb collision and stationary phase approximation
  4. Quantum Coulomb collision with a fixed proton
  5. Numerical sign problem: Fundamental obstacle in negative particle method
  6. Particle generation and variance reduction
  7. Demonstration of the numerical sign problem
  8. Searching for an appropriate filter $\lambda_0$ in SPA
  9. Particle annihilation: A remedy for the numerical sign problem
  10. PAUM: Particle annihilation via uniform mesh
  11. SPADE: An adaptive particle annihilation
  12. Demonstration of SPADE
  13. Bounds for the partition level
  14. Particle annihilation outside domain
  15. Particle simulations of 6-D Wigner-Coulomb dynamics
  16. ...and 29 more sections

Key Result

Theorem 1

\newlabelthm_10 For finite $P$ positive particles $\mathcal{S}^+$ and $M$ negative particles $\mathcal{S}^-$, suppose all overlapped particles carrying opposite sign have been removed, and each bin $\mathrm{Q}_k$ in $\mathcal{P}$ ceases to be split when either the discrepancy bounds def.discrepanc where $D(S^+, S^-) = \sup_{\bm{v} \in [\bm{a}, \bm{b}]} | \frac{1}{P} \sum_{i=1}^P \mathbbm{1}_{\{S

Figures (38)

  • Figure 1: Numerical sign problem in stochastic Wigner simulations: Stochastic errors grow in time due to the accumulation of negative particle weights. WBRW-SPA is able to suppress the growth of errors as it properly accounts for the decay of $\Psi \textup{DO}$ for large $\bm{k}$. $\lambda_0 \le 2$ may underestimate the contribution of low-frequency components and amplify the asymptotic errors, while too large $\lambda_0$ may fail to cancel out the stochastic trajectories efficiently.
  • Figure 1: A complete flow chart of the stochastic Wigner simulations.
  • Figure 1: The 4-D Morse system: A visualization of sign problem and the exponential growth of errors (stochastic variances).
  • Figure 1: The 4-D Morse system: Snapshots of the reduced Wigner function $W_1(x, k, t)$ (left) and the spatial marginal distribution $P(x_1, x_2, t)$ (right) produced by the highly accurate deterministic Wigner solver. \newlabelsupp_snap_asm0
  • Figure 1: Electron-proton interaction: Snapshots of the reduced Wigner functions on $(x_1$-$k_1)$ plane (left) and on $(x_2$-$k_2)$ plane (middle), the spatial marginal distribution (right) produced by the deterministic characteristic-spectral-mixed scheme. \newlabelsupp_qcs_time_evolution0
  • ...and 33 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Proof 1
  • Remark 1
  • Example 1
  • Remark 2
  • Example 2
  • Example 3
  • Theorem 2
  • Proof 2