  In the hydrodynamics of integrable models, diffusion is a subleading
correction to ballistic propagation. Here we quantify the diffusive
contribution for one-dimensional Bose gases and find it most influential in the
crossover between the main thermodynamic regimes of the gas. Analysing the
experimentally measured dynamics of a single density mode, we find diffusion to
be relevant only for high wavelength excitations. Instead, the observed
relaxation is solely caused by a ballistically driven dephasing process, whose
time scale is related to the phonon lifetime of the system and is thus useful
to evaluate the applicability of the phonon bases typically used in quantum
field simulators.
