A series expansion for the time autocorrelation of dynamical variables

TL;DR

This paper introduces a formal series expansion for the time autocorrelation of dynamical variables in Hamiltonian systems and provides criteria to determine the decay rate of correlations, with numerical application to the Fermi-Pasta-Ulam system.

Contribution

It presents a new iterative formula for autocorrelation series expansion and criteria to assess correlation decay rates in Hamiltonian systems.

Findings

Numerical criteria suggest sub-exponential decay in the Fermi-Pasta-Ulam system.

The series expansion applies broadly to Hamiltonian systems with invariant measures.

Indications of slow correlation decay could impact understanding of thermalization.

Abstract

We present here a general iterative formula which gives a (formal) series expansion for the time autocorrelation of smooth dynamical variables, for all Hamiltonian systems endowed with an invariant measure. We add some criteria, theoretical in nature, which enable one to decide whether the decay of the correlations is exponentially fast or not. One of these criteria is implemented numerically for the case of the Fermi-Pasta-Ulam system, and we find indications which might suggest a sub-exponential decay of the time autocorrelation of a relevant dynamical variable.