  We analyze the onset of diffusive hydrodynamics in the one-dimensional
hard-rod gas subject to stochastic backscattering. While this perturbation
breaks integrability and leads to a crossover from ballistic to diffusive
transport, it preserves infinitely many conserved quantities corresponding to
even moments of the velocity distribution of the gas. In the limit of small
noise, we derive the exact expressions for the diffusion and structure factor
matrices, and show that they generically have off-diagonal components in the
presence of interactions. We find that the particle density structure factor is
non-Gaussian and singular near the origin, with a return probability showing
logarithmic deviations from diffusion.
