Correlations of correlations: Secondary autocorrelations in finite harmonic systems

TL;DR

This paper investigates secondary autocorrelations in finite harmonic systems, revealing conditions under which they mirror primary correlations and analyzing how particle mass influences long-term correlation patterns.

Contribution

It introduces a new autocorrelation function of the long-time tail of C(t) and explores its relationship with the primary correlation in finite harmonic systems.

Findings

Secondary autocorrelation can match primary autocorrelation in same-mass systems.

Heavier tagged particles lead to non-random long-time correlation patterns.

Higher order correlations tend to converge to the lowest normal mode.

Abstract

The momentum or velocity autocorrelation function C(t) for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of C(t) ("a correlation of the correlation"), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation C(t) (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations shows non-random long-time scale pattern, and higher order correlations converge to the lowest normal mode.