  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.
