Microscopic and hydrodynamic correlation in 1d hard rod gas

TL;DR

This paper analytically computes microscopic and macroscopic mass density correlations in a 1D hard rod gas, validating hydrodynamic predictions and revealing long-range correlations through exact and BMFT-based calculations.

Contribution

It provides the first exact microscopic correlation calculations and confirms the accuracy of BMFT in describing macroscopic correlations in a 1D hard rod gas.

Findings

Exact microscopic correlation functions derived.

Macroscopic correlations exhibit long-range behavior.

BMFT predictions match systematic coarse-graining results.

Abstract

We compute mass density correlations of a one-dimensional gas of hard rods at both microscopic and macroscopic scales. We provide exact analytical calculations of the microscopic correlation. For the correlation at macroscopic scale,, we utilize Ballistic Macroscopic Fluctuation Theory (BMFT) to derive an explicit expression for the correlations of a coarse-grained mass density, which reveals the emergence of long-range correlations on the Euler space-time scale. By performing a systematic coarse-graining of our exact microscopic results, we establish a micro-macro correspondence and demonstrate that the resulting macroscopic correlations agree precisely with the predictions of BMFT. This analytical verification provides a concrete validation of the underlying assumptions of hydrodynamic theory in the context of hard rod gas.