# Fluctuations in ballistic transport from Euler hydrodynamics

## TL;DR

This paper introduces a formalism based on large deviation theory to exactly analyze fluctuations of conserved quantities in systems described by Euler hydrodynamics, applicable to classical and quantum, integrable or not, in or out of equilibrium.

## Contribution

It provides a universal method to compute full counting statistics for conserved quantities in hydrodynamic systems, linking fluctuation relations to the linearity of hydrodynamics.

## Key findings

- Exact scaled cumulant generating functions derived
- Extended fluctuation relations linked to hydrodynamic linearity
- Phase transition observed in energy transport at sound velocity

## Abstract

We propose a general formalism, within large deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the "extended fluctuation relations" of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of "freeness" much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.