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arXiv · 2602.17963

Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems

Xinyu Liu·Yong Li
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Abstract

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 $\langle G\rangle_t$ from its angular average $\langle \left\langle G \right\rangle_{\theta}\rangle_{0}$ 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 $1/t$ mixing constant $C_G$, a high-frequency Fourier tail, and an explicit normal-form remainder error.

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Xinyu Liu, Yong Li. 2026-02-20. Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems. https://arxiv.org/abs/2602.17963

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