Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems

Xinyu Liu, Yong Li

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 $\langle \left\langle G \right\rangle_θ\rangle_{0}$ 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.

Statistical Ensemble Deviation Estimates for Nearly Integrable Hamiltonian Systems

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 , an explicit upper bound on the deviation of the ensemble average from its angular average 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 mixing constant , a high-frequency Fourier tail, and an explicit normal-form remainder error.
Paper Structure (6 sections, 93 equations)

This paper contains 6 sections, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Lemmas and quantitative estimates for integrable systems
  5. Deviation of statistical ensembles of nearly integrable Hamiltonian
  6. Conclusion

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof