Asymptotic Statistics of Poincar\'e Recurrences in Hamiltonian Systems with Divided Phase Space

TL;DR

This paper investigates the long-term decay behavior of Poincaré recurrences and correlations in Hamiltonian systems with divided phase space, revealing a universal asymptotic decay rate of 1/τ^3 for chaotic dynamics.

Contribution

It demonstrates, using different methods, that Poincaré recurrences and correlations decay as 1/τ^3 in Hamiltonian chaos, and explains the onset of this behavior at large times.

Findings

Poincaré recurrences decay as 1/τ^3 in the standard map with critical golden curve.

Correlation functions also decay as 1/τ^3 asymptotically.

The 1/τ^3 decay exponent is argued to be universal for chaos borders.

Abstract

By different methods we show that for dynamical chaos in the standard map with critical golden curve the Poincar\'e recurrences P(\tau) and correlations C(\tau) asymptotically decay in time as P ~ C/\tau ~ 1/\tau^3. It is also explained why this asymptotic behavior starts only at very large times. We argue that the same exponent p=3 should be also valid for a general chaos border.