On the Time Average of the Autocorrelation Function in Hamiltonian Dynamics

TL;DR

This paper establishes a rigorous lower bound on the time-average of the autocorrelation function in Hamiltonian dynamics, relating it to conserved quantities and ergodic decompositions, and discusses improvements over previous bounds.

Contribution

It provides a new rigorous lower bound on the autocorrelation function's time-average in Hamiltonian systems, improving upon Mazur and Suzuki's bounds.

Findings

Derived a lower bound based on conserved quantities

Connected autocorrelation time averages to ergodic decompositions

Discussed improvements over previous bounds by Mazur and Suzuki

Abstract

Rigorous lower bound on the time-average of the autocorrelation function of an arbitrary L^1 observable is proven in terms of conserved quantities and ergodic decompositions of the Hamiltonian dynamics. Improvements with respect to the bounds given by Mazur and Suzuki are discussed.