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The spatially discrete to continuous limit in the nonlocal quantum diffusion equation

Daniel Matthes, Eva-Maria Rott

TL;DR

This work develops a spatially discrete, finite-dimensional framework for the non-local quantum drift diffusion model nlQDD, grounding the discretization in a discrete quantum Boltzmann equation with BGK relaxation. The authors prove entropy dissipation and global positivity for the discrete nlQDD, derive a diffusive limit to obtain the discrete nlQDD, and establish convergence of discrete solutions to classical nlQDD solutions on time intervals where positivity persists. They further relate discrete and continuous quantum exponentials, provide rigorous error estimates between discrete and continuous Maxwellians, and show that, in the continuous limit, the discrete model recovers the nlQDD dynamics with convergence of the density and potential. The approach circumvents analytic subtleties of the operator-valued nlQDD by exploiting finite-dimensionality, yielding positivity-preserving schemes and extending existence results beyond near-equilibrium data, with a clear pathway to the continuum limit and quantitative kernel approximations.

Abstract

We propose and analyse a spatial discretization of the non-local Quantum Drift Diffusion (nlQDD) model by Degond, Mèhats and Ringhofer in one space dimension. With our approach, that uses consistently matrices on ${\mathbb C}^N$ instead of operators on $L^2$, we circumvent a variety of analytical subtleties in the analysis of the original nlQDD equation, e.g. related to positivity of densities or to the quantum exponential function. Our starting point is spatially discretized quantum Boltzmann equation with a BGK-type collision kernel, from which we derive the discretized nlQDD model in the diffusive limit. Then we verify that solutions dissipate the von-Neumann entropy, which is a known key property of the original nlQDD, and prove global existence of positive solutions, which seems to be a particular feature of the discretization. Our main result concerns convergence of the scheme: discrete solutions converge -- locally uniformly with respect to space and time -- to classical solutions of the the original nlQDD model on any time interval $[0,T)$ on which the latter remain positive. In particular, this extends the existence theory for nlQDD, that has been established only for initial data close to equilibrium so far.

The spatially discrete to continuous limit in the nonlocal quantum diffusion equation

TL;DR

This work develops a spatially discrete, finite-dimensional framework for the non-local quantum drift diffusion model nlQDD, grounding the discretization in a discrete quantum Boltzmann equation with BGK relaxation. The authors prove entropy dissipation and global positivity for the discrete nlQDD, derive a diffusive limit to obtain the discrete nlQDD, and establish convergence of discrete solutions to classical nlQDD solutions on time intervals where positivity persists. They further relate discrete and continuous quantum exponentials, provide rigorous error estimates between discrete and continuous Maxwellians, and show that, in the continuous limit, the discrete model recovers the nlQDD dynamics with convergence of the density and potential. The approach circumvents analytic subtleties of the operator-valued nlQDD by exploiting finite-dimensionality, yielding positivity-preserving schemes and extending existence results beyond near-equilibrium data, with a clear pathway to the continuum limit and quantitative kernel approximations.

Abstract

We propose and analyse a spatial discretization of the non-local Quantum Drift Diffusion (nlQDD) model by Degond, Mèhats and Ringhofer in one space dimension. With our approach, that uses consistently matrices on instead of operators on , we circumvent a variety of analytical subtleties in the analysis of the original nlQDD equation, e.g. related to positivity of densities or to the quantum exponential function. Our starting point is spatially discretized quantum Boltzmann equation with a BGK-type collision kernel, from which we derive the discretized nlQDD model in the diffusive limit. Then we verify that solutions dissipate the von-Neumann entropy, which is a known key property of the original nlQDD, and prove global existence of positive solutions, which seems to be a particular feature of the discretization. Our main result concerns convergence of the scheme: discrete solutions converge -- locally uniformly with respect to space and time -- to classical solutions of the the original nlQDD model on any time interval on which the latter remain positive. In particular, this extends the existence theory for nlQDD, that has been established only for initial data close to equilibrium so far.

Paper Structure

This paper contains 32 sections, 56 theorems, 241 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The nlQDD model in brief
  3. Spatial discretization and main results
  4. Origin of nlQDD: moment hydrodynamics
  5. Derivation of nlQDD: quantum Boltzmann equation
  6. Previous analytical results and novel ideas
  7. Plan of the paper
  8. Notations and Basics
  9. Functions, spaces, operators
  10. Spatial discretization
  11. Discrete operators, matrices, densities
  12. Difference operators
  13. Discrete quantum Liouville equation and its diffusive limit
  14. Quantum Entropy Functional
  15. The discrete quantum Liouville equation
  16. ...and 17 more sections

Key Result

Theorem 1.1

Fix a discretization $\delta=1/N>0$ and an initial density matrix $R^\text{in}\in\mathfrak{D}^+_\delta$. For each $\varepsilon>0$, let $R^\varepsilon:{\mathbb R}_{\ge0}\to\mathfrak{D}^+_\delta$ be the (global unique) solution to eq:DQL0 with $R^\varepsilon(0)=R^\text{in}$, and let $n^\varepsilon:{\m

Figures (1)

  • Figure 1: Our passage from the discrete collisional quantum Liouville equation to nlQDD: we first perform the diffusive limit, resulting in a spatially discrete nlQDD equation, and then pass to the continuous limit.

Theorems & Definitions (123)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Proposition 3.2: Discrete quantum Maxwellian
  • Definition 3.3
  • Remark 3.4
  • proof : Proof of Proposition \ref{['th:quantmax']}
  • Lemma 3.5
  • proof
  • ...and 113 more