  The conventional theory of hydrodynamics describes the evolution in time of
chaotic many-particle systems from local to global equilibrium. In a quantum
integrable system, local equilibrium is characterized by a local generalized
Gibbs ensemble or equivalently a local distribution of pseudo-momenta. We study
time evolution from local equilibria in such models by solving a certain
kinetic equation, the "Bethe-Boltzmann" equation satisfied by the local
pseudo-momentum density. Explicit comparison with density matrix
renormalization group time evolution of a thermal expansion in the XXZ model
shows that hydrodynamical predictions from smooth initial conditions can be
remarkably accurate, even for small system sizes. Solutions are also obtained
in the Lieb-Liniger model for free expansion into vacuum and collisions between
clouds of particles, which model experiments on ultracold one-dimensional Bose
gases.
