On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions

We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper \cite{Boltzmann} within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each {\it entropy-invariant solution} can be identified as a {\it time-periodic Maxwell-Boltzmann distribution}. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} is nonlinearly stable, including neutral asymptotic stability and Lyapunov stability. Furthermore, the convergence rate is entirely reliant on the thirteen conservation laws and is optimal when compared to the linear scenario.