Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models

TL;DR

This paper develops a hydrodynamic fluctuation theory for one-dimensional many-body systems, revealing how noise covariance is modified by long-range correlations and showing that in integrable models, hydrodynamic noise vanishes, confirming recent conjectures.

Contribution

It introduces a modified Kubo formula for hydrodynamic noise in 1D systems and proves that noise vanishes in integrable models, extending the macroscopic fluctuation theory.

Findings

Hydrodynamic noise covariance is given by a modified Kubo formula.

In systems without shocks, corrections remain ballistic and superdiffusive effects are controlled.

Hydrodynamic noise and bare diffusion vanish in integrable systems.

Abstract

Hydrodynamic noise is the Gaussian process that emerges at larges scales of space and time in many-body systems. It is justified by the central limit theorem, and represents degrees of freedom forgotten when projecting coarse-grained observables onto conserved quantities. It is the basis for fluctuating hydrodynamics, where it appears along with bare diffusion terms related to the noise covariance by the Einstein relation. In one spatial dimension, nonlinearities are relevant and may modify the corrections to ballistic behaviours by superdiffusive effects. But in systems where no shocks appear, such as linearly degenerate and integrable systems, the diffusive scaling of these corrections stays intact. Nevertheless, anomalies remain. We show that in such systems, the noise covariance is given by a modification of the Kubo formula, where effects of ballistic long-range correlations have been projected out, and that nonlinearities are tamed by a point-splitting regularisation. With these ingredients, we obtain a well-defined hydrodynamic fluctuation theory in the ballistic scaling of space-time, as a stochastic PDE. It describes the asymptotic expansion in the inverse variation scale of connected correlation functions, self-consistently organised via a cumulant expansion. The resulting anomalous hydrodynamic equation for average densities takes into account both long-range correlations and bare diffusion, generalising recent results. Despite these anomalies, two-point functions satisfy an ordinary diffusion equation, with diffusion matrix determined by the Kubo formula. In integrable systems, we show that hydrodynamic noise, hence bare diffusion, must vanish, as was conjectured recently, and argue that under an appropriate gauge of the currents, this is true at all orders. Thus the Ballistic Macroscopic Fluctuation Theory give the all-order hydrodynamic theory for integrable models.