Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semiclassical limit of entropies and free energies

Zied Ammari, Michele Correggi, Marco Falconi, Raphaël Gautier

TL;DR

The paper investigates the semiclassical limit of entropies and free energies, establishing that after a universal renormalization, both von Neumann and Wehrl entropies converge to the classical Boltzmann entropy for Gibbs states; it then proves that the Wehrl-based free energy Γ-converges to its classical counterpart, ensuring convergence of minimizers. Using Wick quantization, Weyl and coherent-state machinery, and Wigner measures, the authors connect quantum Gibbs states to classical Gibbs measures, showing partition functions and Husimi densities converge appropriately. The results highlight a robust quantum-classical correspondence at thermal equilibrium, with the classical free energy recovered as the Γ-limit of the quantum Wehrl free energy, and with implications for stability of equilibrium states under semiclassical perturbations. The framework relies on detailed semiclassical estimates, convexity arguments, and measure-theoretic tools to bridge quantum and classical descriptions in a controlled variational setting.

Abstract

Entropy and free energy are central concepts in both statistical physics and information theory, with quantum and classical facets. In mathematics these concepts appear quite often in different contexts (dynamical systems, probability theory, von Neumann algebras, etc.). In this work, we study the von Neumann and Wehrl entropies from the point of view of semiclassical analysis. We first prove the semiclassical convergence of the von Neumann to the Wehrl entropy for quantum Gibbs states (thermal equilibrium), after a suitable renormalization has been taken into account. Then, we show that, in the same limit, the free energy functional defined with the Wehrl entropy $ Γ-$converges to its classical counterpart, so implying convergence of the minima and the associated minimizers.

Semiclassical limit of entropies and free energies

TL;DR

The paper investigates the semiclassical limit of entropies and free energies, establishing that after a universal renormalization, both von Neumann and Wehrl entropies converge to the classical Boltzmann entropy for Gibbs states; it then proves that the Wehrl-based free energy Γ-converges to its classical counterpart, ensuring convergence of minimizers. Using Wick quantization, Weyl and coherent-state machinery, and Wigner measures, the authors connect quantum Gibbs states to classical Gibbs measures, showing partition functions and Husimi densities converge appropriately. The results highlight a robust quantum-classical correspondence at thermal equilibrium, with the classical free energy recovered as the Γ-limit of the quantum Wehrl free energy, and with implications for stability of equilibrium states under semiclassical perturbations. The framework relies on detailed semiclassical estimates, convexity arguments, and measure-theoretic tools to bridge quantum and classical descriptions in a controlled variational setting.

Abstract

Entropy and free energy are central concepts in both statistical physics and information theory, with quantum and classical facets. In mathematics these concepts appear quite often in different contexts (dynamical systems, probability theory, von Neumann algebras, etc.). In this work, we study the von Neumann and Wehrl entropies from the point of view of semiclassical analysis. We first prove the semiclassical convergence of the von Neumann to the Wehrl entropy for quantum Gibbs states (thermal equilibrium), after a suitable renormalization has been taken into account. Then, we show that, in the same limit, the free energy functional defined with the Wehrl entropy converges to its classical counterpart, so implying convergence of the minima and the associated minimizers.
Paper Structure (22 sections, 18 theorems, 142 equations)

This paper contains 22 sections, 18 theorems, 142 equations.

Table of Contents

  1. Introduction and Main Results
  2. Gibbs states
  3. Classical and Quantum Entropies
  4. Classical and Quantum Free Energies
  5. Symbols and Quantization
  6. Convergence of Gibbs Entropies
  7. $\Gamma$-convergence of Free Energies
  8. Semiclassical Analysis
  9. Upper and Lower Symbols
  10. Wigner Measures
  11. Gibbs Entropy Convergence
  12. Convergence of Partition Functions
  13. Convergence of the Quantum Gibbs State in the Sense of Wigner Measures
  14. Convergence of the Gibbs Wehrl Entropy
  15. Lower Bound to the von Neumann entropy
  16. ...and 7 more sections

Key Result

Theorem 1.6

[theorem]thm:convergence Let $h \in \mathcal{S}$ and let $H_\varepsilon = h^{\mathrm{Wick}}$ be its Wick quantization. Let also the Gibbs state $\Gamma_{\beta,\varepsilon} := (Z_{\beta, \varepsilon})^{-1} e^{-\beta H_\varepsilon}$ satisfy the assumption A. Then, Gibbs entropies converge: where $\diamond$ stands either for $\mathrm{vN}$ (von Neumann) or $\mathrm{W}$ (Wehrl).

Theorems & Definitions (35)

  • Definition 1.1: Wick quantization
  • Definition 1.2: anti-Wick quantization
  • Definition 1.3: Symbol class $\mathcal{S}$
  • Remark 1.4: Partition functions
  • Remark 1.5: Role of \ref{['A']}
  • Theorem 1.6: Convergence of entropies
  • Remark 1.7: Entropy rescaling
  • Corollary 1.8: Convergence of the Gibbs free energies
  • Definition 1.9: $(X,\mathcal{T})$
  • Remark 1.10
  • ...and 25 more