The fermionic Boltzmann (Boltzmann-Fermi-Dirac or fermionic Nordheim) equation is a kinetic description of rarefied gases of fermions (e.g. electrons). The setting is similar to the classical Boltzmann equation, with a modification of the collision operator, in order to take into account the Pauli exclusion principle. As a result, the corresponding equilibrium distributions (Fermi distributions) and the relevant entropy (Fermi entropy) do also differ from their classical analogues (Maxwellian distribution and Boltzmann entropy).

Entropy methods are a at the core of quantitative studies on relaxation to equilibrium. For the classical Boltzmann equation, the quantitative decay of the relative entropy to equilibrium is provided by a relationship between the relative entropy to equilibrium and its dissipation in time. These relationships are called "Cercignani’s conjecture-type" inequalities.

In this talk, I present a method of "transfer" of inequalities, which establishes an (almost) equivalence, in terms of entropy inequalities, between the classical and the fermionic Boltzmann cases, hence providing a large class of such results for solutions to the fermionic Boltzmann equation, and therefore, quantitative rates of convergence towards equilibrium. I present an application of this result, in which explicit polynomial convergence to equilibrium is rigorously obtained.