At large scales of space and time, the nonequilibrium dynamics of local
observables in extensive many-body systems is extremely well described by
hydrodynamics. At the Euler scale, it is typically assumed that each mesoscopic
region of the system independently reaches a state of maximal entropy under the
constraints given by the available conservation laws: locally, in each “fluid
cell”, one finds a Gibbs or generalised Gibbs state. Away from phase
transitions maximal entropy states show exponential correlation decay, and
independence of fluid cells might be assumed to subsist during the course of
time evolution. In this manuscript, we show that this picture is incorrect:
regions separated by macroscopic distances develop long-range correlations as
time passes. We provide a universal theoretical framework to exactly evaluate
these correlations, an adaptation of the macroscopic fluctuation theory to the
Euler scale. We verify our exact predictions in a paradigmatic interacting
model of many-body physics, the hard-rod gas, by comparing with numerical
simulations and finding excellent agreement.

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