Conserved densities of hard rods: microscopic to hydrodynamic solutions

Abstract 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 a macroscopic scale can be described by generalized 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 being reduced to the Euler solutions of the hydrodynamics in the thermodynamic limit, the microscopic solutions can also capture the effects of the Navier–Stokes terms and thus go beyond the Euler solutions. We demonstrate this feature from microscopic analysis and from numerical solution of the Navier–Stokes equation 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 is different on the two sides of the interface. We supplement our analytical results with extensive numerical simulations.