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Extending Cercignani's conjecture results from Botzmann to Boltzmann-Fermi-Dirac equation

Thomas Borsoni

Abstract

We establish a connection between the relative Classical entropy and the relative Fermi-Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy-entropy production inequality from one case to the other; therefore providing entropy-entropy production inequalities for the Boltzmann-Fermi-Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csisz{á}r-Kullback-Pinsker inequality to weighted Lp norms, 1 $\le$ p $\le$ 2 and a wide class of entropies.

Extending Cercignani's conjecture results from Botzmann to Boltzmann-Fermi-Dirac equation

Abstract

We establish a connection between the relative Classical entropy and the relative Fermi-Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy-entropy production inequality from one case to the other; therefore providing entropy-entropy production inequalities for the Boltzmann-Fermi-Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csisz{á}r-Kullback-Pinsker inequality to weighted Lp norms, 1 p 2 and a wide class of entropies.
Paper Structure (27 sections, 23 theorems, 274 equations)

This paper contains 27 sections, 23 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann-Fermi-Dirac operator
  3. Macroscopic quantities and equilibrium distribution
  4. Fermi-Dirac entropy
  5. Entropy production
  6. Trend to equilibrium and Cercignani's conjecture
  7. Main results
  8. Comparison of the Fermi-Dirac and Classical relative entropies
  9. Entropy-entropy production inequalities for the Boltzmann-Fermi-Dirac (BFD) equation
  10. About the Csiszár-Kullback-Pinsker inequality
  11. Convergence towards equilibrium
  12. Proof of Theorem \ref{['theorem:FD']} and Proposition \ref{['prop:FDupper']}
  13. Key element of the proof
  14. Continuity at zero
  15. Final proof of Theorem \ref{['theorem:FD']}
  16. ...and 12 more sections

Key Result

Proposition 1

For any $\varepsilon \geq 0$ and nonnegative $f \in L^1_2(\mathbb{R}^3)\setminus \{0\}$ such that $1 - \varepsilon f \geq 0$ and either $\gamma > \frac{2}{5}$ if $\varepsilon > 0$ or $f \in L \log L (\mathbb{R}^3)$ if $\varepsilon=0$, denoting by ${\mathcal{M}_{\varepsilon}^f}$ the Fermi-Dirac (or M It holds that

Theorems & Definitions (47)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Remark 2
  • Theorem 1
  • Proposition 2
  • Remark 3
  • Corollary 3
  • Proposition 4
  • Proposition 5
  • ...and 37 more