# Solvable Hydrodynamics of Quantum Integrable Systems

## TL;DR

This paper develops a hydrodynamic framework for quantum integrable systems using a kinetic equation, demonstrating accurate predictions for dynamics in models like XXZ and Lieb-Liniger, relevant to ultracold atom experiments.

## Contribution

It introduces the Bethe-Boltzmann kinetic equation for quantum integrable systems and validates its predictions against numerical simulations and experimental models.

## Key findings

- Hydrodynamics accurately predicts quantum integrable system dynamics.
- Bethe-Boltzmann equation matches RDMG simulations.
- Models replicate ultracold atom experiments.

## Abstract

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.