Hard rod gas with long-range interactions: Exact predictions for hydrodynamic properties of continuum systems from discrete models

TL;DR

This paper establishes a method to predict the hydrodynamic behavior of continuum hard rod gases with long-range interactions by analyzing their discrete exclusion process counterparts, enabling exact solutions.

Contribution

It introduces a correspondence between discrete exclusion processes and continuum hard rod gases with long-range interactions, allowing for exact hydrodynamic predictions.

Findings

Analytical solutions match simulation results for the $ ext{l}$-TASEP.

The approach applies to systems with long-range interactions.

Short-range interaction cases are also addressed.

Abstract

One-dimensional hard rod gases are explicitly constructed as the limits of discrete systems: exclusion processes involving particles of arbitrary length. Those continuum many-body systems in general do not exhibit the same hydrodynamic properties as the underlying discrete models. Considering as examples a hard rod gas with additional long-range interaction and the generalized asymmetric exclusion process for extended particles ($\ell$-ASEP), it is shown how a correspondence between continuous and discrete systems must be established instead. This opens up a new possibility to exactly predict the hydrodynamic behaviour of this continuum system under Eulerian scaling by solving its discrete counterpart with analytical or numerical tools. As an illustration, simulations of the totally asymmetric exclusion process ($\ell$-TASEP) are compared to analytical solutions of the model and applied to the corresponding hard rod gas. The case of short-range interaction is treated separately.