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The entropic central limit theorem for stochastic integrable Hamiltonian systems

Chen Wang, Yong Li

TL;DR

This paper develops an information-theoretic entropic central limit theorem for stochastic perturbations of finite-dimensional integrable Hamiltonian systems. By analyzing the lattice-valued discretized frequency sequence $\{\hat{\omega}_n\}$ and its partial sums, it shows that the relative entropy to a Gaussian with matched mean and covariance, the Shannon entropy discrepancy corrected for lattice spacing, and the total variation distance all vanish asymptotically, indicating maximal entropy behavior while preserving invariant tori structure. The approach combines a Bernoulli decomposition and Lindeberg-type arguments to handle independent but non-identically distributed components, yielding an entropic refinement of the classical CLT and a rigorous information-theoretic account of how chaotic, yet structured, orbital dynamics persist under stochastic perturbations. The results extend to multi-dimensional settings and admit a corollary that recovers the continuous case as the discretization scale vanishes, linking entropy convergence to the persistence of invariant tori and global coverage of the toroidal phase space. This provides a quantitative, entropy-based lens on the second law of thermodynamics in dynamical systems and broadens the toolkit for studying stochastic Hamiltonian systems.

Abstract

In this paper, we investigate the asymptotic stability of finite-dimensional stochastic integrable Hamiltonian systems via information entropy. Specifically, we establish the asymptotic vanishing of Shannon entropy difference (with correction for the lattice interval length) and relative entropy between the partial sum of discretized frequency sequence and its quantized Gaussian approximation (expectation and covariance variance matched). These two convergence are logically consistent with the second law of thermodynamics: the complexity of the system has reached the theoretical limit, and the orbits achieve a global unbiased coverage of the invariant tori with the most thorough chaotic behavior, their average winding rate along the tori stays fixed at the corresponding expected value of the frequency sequence, while deviations from this average follow isotropic Gaussian dynamics, much like Wiener process around a fixed trajectory. This thus provides an information-theoretic quantification of how orbital complexity ensures the persistence of invariant tori beyond mere convergence of statistical distributions (as stated in the classical central limit theorem).

The entropic central limit theorem for stochastic integrable Hamiltonian systems

TL;DR

This paper develops an information-theoretic entropic central limit theorem for stochastic perturbations of finite-dimensional integrable Hamiltonian systems. By analyzing the lattice-valued discretized frequency sequence and its partial sums, it shows that the relative entropy to a Gaussian with matched mean and covariance, the Shannon entropy discrepancy corrected for lattice spacing, and the total variation distance all vanish asymptotically, indicating maximal entropy behavior while preserving invariant tori structure. The approach combines a Bernoulli decomposition and Lindeberg-type arguments to handle independent but non-identically distributed components, yielding an entropic refinement of the classical CLT and a rigorous information-theoretic account of how chaotic, yet structured, orbital dynamics persist under stochastic perturbations. The results extend to multi-dimensional settings and admit a corollary that recovers the continuous case as the discretization scale vanishes, linking entropy convergence to the persistence of invariant tori and global coverage of the toroidal phase space. This provides a quantitative, entropy-based lens on the second law of thermodynamics in dynamical systems and broadens the toolkit for studying stochastic Hamiltonian systems.

Abstract

In this paper, we investigate the asymptotic stability of finite-dimensional stochastic integrable Hamiltonian systems via information entropy. Specifically, we establish the asymptotic vanishing of Shannon entropy difference (with correction for the lattice interval length) and relative entropy between the partial sum of discretized frequency sequence and its quantized Gaussian approximation (expectation and covariance variance matched). These two convergence are logically consistent with the second law of thermodynamics: the complexity of the system has reached the theoretical limit, and the orbits achieve a global unbiased coverage of the invariant tori with the most thorough chaotic behavior, their average winding rate along the tori stays fixed at the corresponding expected value of the frequency sequence, while deviations from this average follow isotropic Gaussian dynamics, much like Wiener process around a fixed trajectory. This thus provides an information-theoretic quantification of how orbital complexity ensures the persistence of invariant tori beyond mere convergence of statistical distributions (as stated in the classical central limit theorem).
Paper Structure (4 sections, 7 theorems, 96 equations)

This paper contains 4 sections, 7 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result and proof
  4. Proof of Lemma 3.2

Key Result

Lemma 3.1

Suppose that each $\kappa_{i}=0$. Then the corresponding partial sum $S_{n}$ and the standardised sum $S_{n}'$ satisfy that, as $n\to \infty$, where in particular, for any $n\ge 1$, the $O\left(1/s_{n}\right)$ can be absolutely bounded by $l/s_{n}+l^{2}/2s_{n}^{2}$. Moreover, let $U$ be an independent uniform random variable on $(-1/2,1/2)$. Then as $n\to \infty$, where the error term $O\left(1/

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 3.1
  • Proof 3.1
  • Remark 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Proof 3.2
  • Proposition 3.3
  • Proof 3.3
  • Remark 3.2
  • ...and 9 more