[Paper Review] From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

Jacky J. Chong, Laurent Laflèche+1 more|arXiv (Cornell University)|Mar 19, 2021

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper establishes the rigorous derivation of the Vlasov equation with singular potentials from the many-body Schrödinger equation for fermions, via a combined mean-field and semiclassical limit. It proves uniform-in-ℏ propagation of regularity for the Hartree-Fock equation with singular interactions (e.g., Coulomb, gravitational), enabling quantitative estimates in Schatten norms and deriving the Vlasov equation for a ∈ (0, 1/2) on local-in-time scales under the scaling N−1/2 ≪ ℏ ≤ N−1/3.

ABSTRACT

We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.

Motivation & Objective

• To rigorously derive the Vlasov equation with singular pair potentials (e.g., Coulomb, gravitational) from the N-body Schrödinger equation for fermions.
• To establish uniform-in-Planck-constant ℏ propagation of regularity for the Hartree-Fock equation with singular potentials of the form ±|x−y|−a.
• To obtain quantitative estimates on the distance between many-body Schrödinger dynamics and the Hartree-Fock/Vlasov equations in Schatten norms under combined mean-field and semiclassical limits.
• To analyze the validity of the Vlasov approximation in regimes where ℏ ≪ 1 and N is large, with particular attention to the critical exponent a = 1/2 and the role of N-dependent cutoffs for a ≥ 1/2.

Proposed method

• Prove uniform-in-ℏ propagation of regularity for the Hartree-Fock equation with singular potentials |x−y|−a via commutator estimates and functional calculus in semiclassical and fermionic Fock spaces.
• Use second quantization and state purification to reduce the N-body problem to a dynamics on Fock space with quasi-free initial states.
• Apply Bogoliubov transformations and fluctuation dynamics to control quantum fluctuations and derive the mean-field limit.
• Employ Schatten norm estimates to compare solutions of the Schrödinger, Hartree-Fock, and Vlasov equations, with bounds depending on ℏ, N, and the singularity strength a.
• Introduce a decomposition of the dynamics into regular and singular parts, using cutoffs and weighted norms to control divergent terms in the commutator estimates.
• Leverage the structure of the exchange operator and the kernel singularity to derive time-regularity and Hölder continuity of the dynamics in the Schatten norm.

Experimental results

Research questions

• RQ1 Can the Vlasov equation with singular potentials be rigorously derived from the many-body Schrödinger equation in the combined mean-field and semiclassical limit?
• RQ2 What is the optimal scaling regime (in terms of N and ℏ) for the validity of the Hartree-Fock and Vlasov approximations with singular interactions?
• RQ3 How does the strength of the singularity a in |x−y|−a affect the time of validity of the mean-field and semiclassical limits?
• RQ4 What role does the fermionic nature of the particles play in the regularity and convergence of the Hartree-Fock dynamics?
• RQ5 Can uniform-in-ℏ regularity be established for the Hartree-Fock equation with singular pair potentials, and how does this enable quantitative convergence estimates?

Key findings

• For interaction potentials with a ∈ (0, 1/2), the Vlasov equation is derived in a local-in-time regime under the scaling N−1/2 ≪ ℏ ≤ N−1/3.
• Uniform-in-ℏ propagation of regularity is established for the Hartree-Fock equation with singular potentials |x−y|−a for all a ∈ (0,1), enabling quantitative convergence estimates.
• The distance between solutions of the many-body Schrödinger equation and the Hartree-Fock equation is bounded in Schatten norms by O(ℏ R^{-(a+2)}) and O(ℏ R^{3/2 - a}) for regular and singular parts, respectively.
• For a ∈ [1/2, 1], the convergence holds only on a small time scale or with an N-dependent cutoff, indicating a breakdown of the approximation for stronger singularities.
• The proof relies on detailed commutator estimates involving the direct and exchange terms in the Hartree-Fock Hamiltonian, with careful control of singular kernels via weighted norms and Hölder continuity.
• The dynamics of the fluctuation operator are shown to be Hölder continuous in time, which is essential for the convergence argument in Schatten norms.