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On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm

Francis Filbet, François Golse

TL;DR

The paper develops an asymptotic-preserving numerical framework for the von Neumann equation in the semiclassical limit by reformulating the problem in Weyl variables to remove stiffness and applying a truncated Hermite expansion in the auxiliary variable. The resulting Hermite-Galerkin system in $y$ is coupled with a finite-volume discretization in $x$ and Crank–Nicolson time stepping, with a tailored fast approximation of the nonlocal term for $\hbar\ll 1$. The authors provide error estimates uniform in $\hbar$, demonstrate $L^2$ stability, and validate the method through numerical experiments on harmonic, quartic, barrier, and Morse potentials, capturing quantum revivals, tunneling, and phase-space dynamics across regimes. The approach preserves key quantum-state properties and offers substantial efficiency gains for mixed states with smooth Wigner functions, with potential extensions to stochastic schemes. Overall, the work contributes a robust, scalable tool for simulating the von Neumann equation in both quantum and semiclassical contexts.

Abstract

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl's variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not.

On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm

TL;DR

The paper develops an asymptotic-preserving numerical framework for the von Neumann equation in the semiclassical limit by reformulating the problem in Weyl variables to remove stiffness and applying a truncated Hermite expansion in the auxiliary variable. The resulting Hermite-Galerkin system in is coupled with a finite-volume discretization in and Crank–Nicolson time stepping, with a tailored fast approximation of the nonlocal term for . The authors provide error estimates uniform in , demonstrate stability, and validate the method through numerical experiments on harmonic, quartic, barrier, and Morse potentials, capturing quantum revivals, tunneling, and phase-space dynamics across regimes. The approach preserves key quantum-state properties and offers substantial efficiency gains for mixed states with smooth Wigner functions, with potential extensions to stochastic schemes. Overall, the work contributes a robust, scalable tool for simulating the von Neumann equation in both quantum and semiclassical contexts.

Abstract

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl's variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not.
Paper Structure (15 sections, 2 theorems, 88 equations, 12 figures, 1 table)

This paper contains 15 sections, 2 theorems, 88 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum dynamics in Weyl's variables
  3. Semi-classical limit
  4. Hermite spectral method
  5. Hermite approximation of the von Neumann equation
  6. Review of error estimates
  7. Finite volume in $x$ and time discretization
  8. Finite volume method
  9. Approximation of the non-local term when $\hbar\ll 1$
  10. Numerical simulations
  11. Harmonic potential
  12. Quantum revivals
  13. Quantum tunneling
  14. The Morse potential
  15. Conclusion

Key Result

Theorem 1.1

Assume that $V$ satisfies hyp:V1-hyp:V2 with $m=3$ while for all integers $a$, $b$, $\alpha$ and $\beta$ such that $a+b+\alpha+\beta\le 3$, we suppose that both initial data $R^{in}_\hbar$ and $\hat{R}^{in}_\hbar$ are such that where $\mathcal{C}_0>0$ does not depend on $\hbar$ and $L^2=L^2_{xy}:=L^2({\mathbb R}^2)$. Then the solution $R$ to the von Neumann equation vNvW and the solution $\hat{R}

Figures (12)

  • Figure 4.1: Quantum revivals : Time evolution of the kinetic energy $\mathcal{K}(t)$ for different values of $\hbar=0.01$, $\hbar=0.1$, $\hbar=0.2$ and $\hbar=0.5$.
  • Figure 4.2: Quantum revivals : Snapshots of the Wigner distribution $W(t,x,\xi)$ at time $t=0$, $10$, $20$, $30$, $50$ and $70$ for $\hbar=0.01$.
  • Figure 4.3: Quantum revivals : Snapshots of the Wigner distribution $W(t,x,\xi)$ at time $t=0$, $10$, $20$, $30$, $50$ and $70$ for $\hbar=0.5$.
  • Figure 4.4: Quantum revivals : Snapshots of the matrix density $R(t,x,y)$ and the density $\rho$, momentum $\rho \;u$ and energy density $\rho\, e$ at time $t=50$ for $\hbar=0.5$ (top) and $\hbar=0.01$ (bottom).
  • Figure 4.5: Quantum tunneling : Time evolution of the Wigner function computed from the approximation of $R$ with $\hbar=0.1$. The lines represent the level set of the Hamiltonian $H(x,\xi) = V(x) + \xi^2/2$.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1