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Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

François Golse, Norbert J. Mauser, Jakob Möller

TL;DR

Addresses whether velocity averaging lemmas extend from classical kinetic equations to quantum (Wigner) and semi-classical regimes. The authors formulate a quantum averaging lemma for Wigner functions at fixed ħ, yielding $H^{1/2}$ regularity for velocity averages under a mixed-state occupation bound, with bounds deteriorating as ħ→0. They further establish a ħ-uniform semi-classical averaging lemma for mixed states, obtaining $H^{1/4}$ regularity, while showing that pure states can be monokinetic in the semiclassical limit, precluding general averaging in that regime. The work links quantum transport to Quantum Hydrodynamics via Madelung-type derivations, clarifying when velocity averaging can inform semi-classical limits and hydrodynamic models. Overall, it extends velocity averaging techniques to quantum contexts and clarifies their limitations in pure-state and semi-classical settings.

Abstract

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space $(x,v)$ containing a transport ("advection") term. By integrating over $v$ in velocity space $\mathbb{R}_v^d$ (velocity averaging), one gains regularity for the density in position space $\mathbb{R}_x^d$. The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in "classical kinetic theory". In "quantum kinetic theory", the Schrödinger equation for the complex-valued "wave function" in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ("Quantum Vlasov") equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the "quantum" case of the Wigner equation. The hard part are the "semi-classical" averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context "pure vs. mixed states" play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the "semi-classical" case which is worked out in full detail in a follow-up article.

Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

TL;DR

Addresses whether velocity averaging lemmas extend from classical kinetic equations to quantum (Wigner) and semi-classical regimes. The authors formulate a quantum averaging lemma for Wigner functions at fixed ħ, yielding regularity for velocity averages under a mixed-state occupation bound, with bounds deteriorating as ħ→0. They further establish a ħ-uniform semi-classical averaging lemma for mixed states, obtaining regularity, while showing that pure states can be monokinetic in the semiclassical limit, precluding general averaging in that regime. The work links quantum transport to Quantum Hydrodynamics via Madelung-type derivations, clarifying when velocity averaging can inform semi-classical limits and hydrodynamic models. Overall, it extends velocity averaging techniques to quantum contexts and clarifies their limitations in pure-state and semi-classical settings.

Abstract

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space containing a transport ("advection") term. By integrating over in velocity space (velocity averaging), one gains regularity for the density in position space . The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in "classical kinetic theory". In "quantum kinetic theory", the Schrödinger equation for the complex-valued "wave function" in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ("Quantum Vlasov") equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the "quantum" case of the Wigner equation. The hard part are the "semi-classical" averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context "pure vs. mixed states" play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the "semi-classical" case which is worked out in full detail in a follow-up article.

Paper Structure

This paper contains 9 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Classical averaging lemmas
  3. Quantum averaging lemmas
  4. Preliminaries
  5. Velocity Averaging for the "quantum kinetic" Wigner equation
  6. Semi-classical averaging lemmas
  7. Semi-classical velocity averaging for mixed states
  8. Semi-classical velocity averaging for pure states
  9. Connection to Quantum Hydrodynamics

Key Result

Theorem 1

Let $n\ge 0$ and $f \in L^2(\mathbb{R}_t\times \mathbb{R}^d_x \times \mathbb{R}^d_{v})$ satisfy where $S[f] \in L^2(\mathbb{R}_t \times \mathbb{R}^d_x, H^{-n}(\mathbb{R}^d_{v}))$. Then for each $\psi\in\mathcal{S}(\mathbb{R}^d_v)$,

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 1
  • Corollary 1
  • Proposition 1
  • Remark 2