  Obtaining rigorous and general results about the non-equilibrium dynamics of
extended many-body systems is a difficult task. In quantum lattice models with
short-range interactions, the Lieb-Robinson bound tells us that the spatial
extent of operators grows at most linearly in time. But what happens within
this light-cone? We discuss rigorous results on ergodicity and the emergence of
the hydrodynamic scale, which establish fundamental principles at the root of
non-equilibrium physics. One key idea of the present work is that general
structures of hydrodynamics at the Euler scale follow independently from the
details of the microscopic dynamics, and in particular do not necessitate
chaos; they are consequences of "extensivity". Another crucial observation is
that these apply at arbitrary frequencies and wavelengths. That is, long-time
oscillatory behaviours can be reproduced from a natural extension of standard
hydrodynamic notions, thus enlarging the hydrodynamic paradigm beyond the
zero-frequency / infinite-wavelength point that it traditionally addresses.
