Correlation Functions for a Chain of Short Range Oscillators

TL;DR

This paper analyzes the asymptotic behavior of correlation functions in a chain of harmonic oscillators with short-range interactions, revealing universal peak shapes and decay rates, and explores effects of nonlinear perturbations.

Contribution

It introduces a generalized spacings approach for correlation analysis and provides a detailed asymptotic description of correlation peaks using Airy and Pearcey functions.

Findings

Fastest peaks move in opposite directions with decay rates $t^{-1/3}$ and $t^{-2/3}$.

Additional non-generic peaks decay at different rates, described by Pearcey integrals.

Numerical simulations show nonlinearities affect peak shapes and decay rates.

Abstract

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate $t^{-\frac{1}{3}}$ for position and momentum correlations and as $t^{-\frac{2}{3}}$ for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate $t^{-\frac{1}{4}}$ for position and momentum correlators and with rate $t^{-\frac{1}{2}}$ for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.