Equilibrium dynamical correlations in the Toda chain and other integrable models

TL;DR

This paper studies equilibrium spatio-temporal correlations and energy transport in the Toda lattice and other integrable models, revealing ballistic scaling and exact correlations in certain limits, with implications for understanding integrable systems.

Contribution

It demonstrates that equilibrium correlations in integrable models exhibit ballistic scaling and provides exact results in specific limiting cases, enhancing understanding of integrable dynamics.

Findings

Correlations satisfy ballistic scaling with data collapse over time

Exact correlation functions obtained in harmonic and hard-particle gas limits

Normal mode transformation is useful even for integrable systems

Abstract

We investigate the form of equilibrium spatio-temporal correlation functions of conserved quantities, and of energy transport in the Toda lattice and in other integrable models. From numerical simulations we find that the correlations satisfy ballistic scaling with a remarkable collapse of data from different times. We examine special limiting choices of parameter values, for which the Toda lattice tends to either the harmonic chain or the equal mass hard-particle gas. In both these limiting cases, one can obtain the correlations exactly and we find excellent agreement with the direct Toda simulation results. We also discuss a transformation to "normal mode" variables, as commonly done in hydrodynamic theory of non-integrable systems, and find that this is useful, to some extent, even for the integrable system.