  In the hydrodynamic theory, the non-equilibrium dynamics of a many-body
system is approximated, at large scales of space and time, by irreversible
relaxation to local entropy maximisation. This results in a convective equation
corrected by viscous or diffusive terms in a gradient expansion, such as the
Navier-Stokes equations. Diffusive terms are evaluated using the Kubo formula,
and possibly arising from an emergent noise due to discarded microscopic
degrees of freedom. In one dimension of space, diffusive scaling is often
broken as noise leads to super-diffusion. But in linearly degenerate
hydrodynamics, such as that of integrable models, diffusive behaviors are
observed, and it has long be thought that the standard diffusive picture
remains valid. In this letter, we show that in such systems, the Navier-Stokes
equation breaks down beyond linear response. We demonstrate that
diffusive-order corrections do not take the form of a gradient expansion.
Instead, they are completely determined by ballistic transport of initial-state
fluctuations, and obtained from the non-local two-point correlations recently
predicted by the ballistic macroscopic fluctuation theory (BMFT); the resulting
hydrodynamic equations are reversible. To do so, we establish a regularised
fluctuation theory, putting on a firm basis the recent idea that ballistic
transport of initial-state fluctuations determines fluctuations and
correlations beyond the Euler scale. This extends the idea of "diffusion from
convection" previously developed to explain the Kubo formula in integrable
systems, to generic non-equilibrium settings.
