  We consider a system of many hard rods moving in one-dimension. As it is an
integrable system, it possesses an extensive number of conserved quantities and
its evolution on macroscopic scale can be described by generalised
hydrodynamics. Using a microscopic approach, we compute the evolution of the
conserved densities starting from non-equilibrium initial conditions of both
quenched and annealed type. In addition to getting reduced to the Euler
solutions of the hydrodynamics in the thermodynamic limit, the microscopic
solutions can also capture effects of the Navier-Stokes terms and thus go
beyond the Euler solutions. We demonstrate this feature in two problems -
first, tracer diffusion in a background of hard rods and second, the evolution
from a domain wall initial condition in which the velocity distribution of the
rods are different on the two sides of the interface. We supplement our
analytical results using extensive numerical simulations.
