Quantitative relaxation towards equilibrium for solutions to the Boltzmann-Fermi-Dirac equation with cutoff hard potentials (Q6592094)

In this paper, the authors provide the first quantitative result on the convergence rate of the solutions to the spatially homogeneous quantum Boltzmann equation for hard potentials with cutoff towards the quantum equilibrium state. The authors consider the model for Fermi-Dirac statistics. For each fixed Planck constant \(\varepsilon>0\), it has been shown in [J. Stat. Phys. 190, No. 8, Paper No. 139, 36 p. (2023; Zbl 07745915)] by \textit{B. Liu} and \textit{X. Lu} that a solution to the spatially homogeneous quantum Boltzmann equation for fermions converges to the Fermi-Dirac equilibrium state in \(L^1_2\) at \(t\to \infty\). But the explicit convergence rate has not been known, and the authors in this paper prove that, for hard potentials with angular cutoff, the relative entropy decays polynomially as \((1+t)^{-1/\gamma}\) where \(\gamma\) is the exponent for the kinetic part of the Boltzmann cross-section. In particular, for any \(p\ge 1\) and \(k\ge 0,\) the authors prove that the difference of the solution and the equilibrium in \(L^p_k\) decays as \((1+t)^{-1/(2p\gamma)}\).