Statistical ensembles in integrable Hamiltonian systems with almost periodic transitions

We study the long-term average evolution of the random ensemble along integrable Hamiltonian systems with time $T$-periodic transitions. More precisely, for any observable $G$, it is demonstrated that the ensemble under $G$ in long time average converges to that over one time period $T$, and that the probability measure induced by the probability density function describing the ensemble at time $t$ weakly converges to the average of the probability measures over time $T$. And we extend the result to almost periodic cases. The key to the proof is based on the {\it {Riemann-Lebesgue lemma in time-average form}} generalized in the paper. %This work contributes to the comprehension of the statistical mechanics of Hamiltonian systems subject to disturbances.