When a gas of particles interacts with much a larger reservoir the density dynamics on large scales is typically governed by diffusion. We study this paradigmatic problem for weakly coupled integrable systems and show that this picture gets altered when transport is investigated on top of long-lived nonthermal states. Remarkably, for states noninvariant under parity we find the Burgers equation arising in the hydrodynamic limit. We explicitly compute the diffusion constant and nonlinear advective coefficient of the Burgers equation using a variant of the Chapman-Enskog theory. We find an excellent agreement between our theory and numerical simulations of a simplified model of stochastic two-body collisions. Our conclusions are based only on Galilean invariance, existence of a small system-bath coupling parameter, and a small momentum exchange between the system and the bath particles during two-body scattering.

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• Kinetic theory
• Ultracold gases
• Hydrodynamics
• Integrable systems