Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems

TL;DR

This paper derives explicit, long-time deviation bounds for statistical ensembles in nearly integrable Hamiltonian systems by combining stability estimates with phase-mixing techniques, providing detailed error contributions.

Contribution

It introduces a novel method combining Nekhoroshev stability with phase-mixing to quantify ensemble deviations over exponentially long times.

Findings

Explicit deviation bounds for ensemble averages over long times

Separation of resonant and nonresonant contributions

Quantitative error estimates involving normal-form remainders

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.