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.
Benjamin Doyon (Thu,) studied this question.