The entropic central limit theorem for stochastic integrable 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).