Dean-Kawasaki fluctuating hydrodynamics for backscattering hard rods

We study a system of backscattering hard rods in one dimension. Contrary to the usual ballistic hard rods, these hard rods flip the sign of their velocities with a rate $\gamma$. This leads to the decay of the odd moments of velocity while preserving the even moments: the number of conserved quantities in the system becomes half. The introduction of the flipping rate $\gamma$ is an integrability-breaking perturbation, and this leads to a change in the transport properties in the system. We show using a Dean-Kawasaki fluctuating hydrodynamic formulation that the unequal space-time correlation of the normal mode phase space densities attains a diffusive form at late times. Also, we show that for $t \gg 1/\gamma$, the two-time density-density correlation of mass densities spreads in a diffusive manner, and for $t \ll 1/\gamma$, the correlation spreads ballistically, for a background state given by the Boltzmann distribution. Our results present an elegant framework to study systems where integrability is broken by a stochastic noise.