On universality of algebraic decays in Hamiltonian systems

TL;DR

This paper proposes a universal algebraic decay law for correlations and recurrences in Hamiltonian systems, supported by a Markov tree model and numerical simulations across various systems.

Contribution

It introduces a conjecture of universal asymptotic decay in Hamiltonian systems, backed by a Markov model and numerical evidence.

Findings

Numerical simulations support the universality conjecture.

A Markov tree model predicts a universal decay exponent.

Finite-time experiments show system-dependent exponents, but asymptotic behavior is universal.

Abstract

Hamiltonian systems with a mixed phase space typically exhibit an algebraic decay of correlations and of Poincare' recurrences, with numerical experiments over finite times showing system-dependent power-law exponents. We conjecture the existence of a universal asymptotic decay based on results for a Markov tree model with random scaling factors for the transition probabilities. Numerical simulations for different Hamiltonian systems support this conjecture and permit the determination of the universal exponent.