  We consider the relaxation of an initial non-equilibrium state in a
one-dimensional fluid of hard rods. Since it is an interacting integrable
system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long
times for generic initial conditions. Here we show that there exist initial
conditions for which the system does not reach GGE even at very long times and
in the thermodynamic limit. In particular, we consider an initial condition of
uniformly distributed hard-rods in a box with the left half having particles
with a singular velocity distribution (all moving with unit velocity) and the
right half particles in thermal equilibrium. We find that the density profile
for the singular component does not spread to the full extent of the box and
keeps moving with a fixed effective speed at long times. We show that such
density profiles can be well described by the solution of the Euler equations
almost everywhere except at the location of the shocks, where we observe slight
discrepancies due to dissipation arising from the initial fluctuations of the
thermal background. To demonstrate this effect of dissipation analytically, we
consider a second initial condition with a single particle at the origin with
unit velocity in a thermal background. We find that the probability
distribution of the position of the unit velocity quasi-particle has diffusive
spreading which can be understood from the solution of the Navier-Stokes
equation of the hard rods. Finally, we consider an initial condition with a
spread in velocity distribution for which we show convergence to GGE. Our
conclusions are based on molecular dynamics simulations supported by analytical
arguments.
