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On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation

Jacky J. Chong, Laurent Lafleche, Chiara Saffirio

TL;DR

The paper rigorously characterizes the semiclassical limit $\hbar\to 0$ of the Hartree equation with Coulomb interactions, showing that the Wigner transform of the Hartree solution converges in $L^2$ to a Vlasov–Poisson solution at a rate linear in $\hbar$. A novel stability framework for the square root of phase-space densities is developed, together with Wick quantization to overcome positivity issues and obtain explicit, uniform-in-$\hbar$ bounds. The analysis introduces quantum Sobolev-type spaces and detailed commutator estimates, enabling a precise, quantitative comparison between quantum and classical evolutions in the Schatten norm setting. The results improve previous rates and offer a method that parallels mean-field derivations for mixed quantum states, with potential insights for the quantum-to-classical transition in many-body dynamics. The work thus provides a sharp, practically applicable link between quantum Hartree dynamics and classical Vlasov–Poisson dynamics in the singular Coulomb regime.

Abstract

Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states.

On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation

TL;DR

The paper rigorously characterizes the semiclassical limit of the Hartree equation with Coulomb interactions, showing that the Wigner transform of the Hartree solution converges in to a Vlasov–Poisson solution at a rate linear in . A novel stability framework for the square root of phase-space densities is developed, together with Wick quantization to overcome positivity issues and obtain explicit, uniform-in- bounds. The analysis introduces quantum Sobolev-type spaces and detailed commutator estimates, enabling a precise, quantitative comparison between quantum and classical evolutions in the Schatten norm setting. The results improve previous rates and offer a method that parallels mean-field derivations for mixed quantum states, with potential insights for the quantum-to-classical transition in many-body dynamics. The work thus provides a sharp, practically applicable link between quantum Hartree dynamics and classical Vlasov–Poisson dynamics in the singular Coulomb regime.

Abstract

Using a new stability estimate for the difference of the square roots of two solutions of the VlasovPoisson equation, we obtain the convergence in the norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the VlasovPoisson equation, with a rate of convergence proportional to . This improves the rate of convergence in obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the HartreeFock equation for mixed states.
Paper Structure (20 sections, 13 theorems, 156 equations)

This paper contains 20 sections, 13 theorems, 156 equations.

Table of Contents

  1. Introduction and main result
  2. Notations
  3. Functional spaces
  4. Weyl quantization and Wigner transform
  5. Main result
  6. Strategy and explicit constants
  7. Quantum Sobolev spaces
  8. A precised version of the main theorem
  9. Stability estimates for the Vlasov--Poisson and Hartree equations
  10. Classical case
  11. Quantum case
  12. A Semiclassical inequality for commutators
  13. Semiclassical limit in L2 Schatten norm
  14. An estimate on the lack of positivity
  15. Trace norm estimate.
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\boldsymbol{\rho}\geq 0$ be a solution to the Hartree equation eq:Hartree with initial condition $\boldsymbol{\rho}^\mathrm{in}\in\mathcal{L}^1\cap\mathcal{L}^\infty$ and $f\geq 0$ be a solution to the Vlasov--Poisson equation eq:Vlasov with initial condition $f^\mathrm{in}$ verifying for $n_1 > 6$. Then there exist $\Lambda, C_1, C_2 \in C^0(\mathbb R_+,\mathbb R_+)$, independent of $\hbar$

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • ...and 21 more