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On local quantum Gibbs states

Romain Duboscq, Olivier Pinaud

Abstract

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose macroscopic quantities such as the particle density, current, and kinetic energy are fixed at each point of $\Rm^d$, and look for a density operator over $L^2(\Rm^d)$ minimizing an entropy functional. Such minimizers are referred to as a local Gibbs states. This setting is in constrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises for instance in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.

On local quantum Gibbs states

Abstract

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose macroscopic quantities such as the particle density, current, and kinetic energy are fixed at each point of , and look for a density operator over minimizing an entropy functional. Such minimizers are referred to as a local Gibbs states. This setting is in constrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises for instance in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.

Paper Structure

This paper contains 28 sections, 13 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Main results
  4. Functional setting
  5. The constraints
  6. Results
  7. Strategy of the proof
  8. Proof of Theorem \ref{['mainth']}
  9. Preliminary technical lemmas
  10. Step 1: reduction to the energy constraint
  11. Step 2: reduction to global constraints
  12. Step 3: the global minimization problem
  13. Proof of Proposition \ref{['exglob']}, step 1: an auxiliary problem.
  14. Proof of Proposition \ref{['exglob']}, step 2: monotonicity argument.
  15. Step 4: Conclusion
  16. ...and 13 more sections

Key Result

Theorem \oldthetheorem

Under Assumptions A0-A1-A2-A3-A4, the entropy $S$ admits a unique minimizer in $\mathcal{A}(n_0,u_0,k_0)$.

Theorems & Definitions (24)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 14 more