Extending Cercignani’s Conjecture Results from Boltzmann 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 Csiszár–Kullback–Pinsker inequality to weighted \(L^p\) norms, \(1 \le p \le 2\) and a wide class of entropies.