Conserved densities of hard rods: microscopic to hydrodynamic solutions

TL;DR

This paper analyzes the microscopic and hydrodynamic evolution of conserved densities in a one-dimensional system of hard rods, demonstrating how microscopic solutions can incorporate effects beyond classical hydrodynamics.

Contribution

It provides a microscopic derivation of conserved density evolution in hard rods, capturing both Euler and Navier-Stokes effects through analytical and numerical methods.

Findings

Microscopic solutions match hydrodynamic Euler solutions in the thermodynamic limit.

Microscopic analysis captures Navier-Stokes effects such as diffusion.

Numerical simulations confirm the analytical predictions for tracer diffusion and domain wall evolution.

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 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 from microscopic analysis and 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 are different on the two sides of the interface. We supplement our analytical results using extensive numerical simulations.