Emergent hydrodynamics in integrable systems out of equilibrium

The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale, the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases, and classical ones such as soliton gases. I will give an overview of what GHD is, its relation to quantum integrable systems and to gases of classical solitons, how it leads to exact results in transport problems, and if time permits some geometric ideas underlying it.