Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems

This paper studies quantitative deviation bounds for statistical ensembles evolving under the one-parameter flow of a nearly integrable Hamiltonian system. Combining Nekhoroshev-type stability estimates with phase-mixing arguments, we obtain, for any observable $G$, an explicit upper bound on the deviation of the ensemble average $⟨G{⟩}_t$ from its angular average $⟨{⟨G⟩}_{\theta }{⟩}_0$ over exponentially long time scales. The bound separates contributions from the resonant neighborhood via a probability-mass term, and from the nonresonant region via a traceable $1\mathrm{/}t$ mixing constant $C_G$, a high-frequency Fourier tail, and an explicit normal-form remainder error.