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Microscopic and hydrodynamic correlation in 1d hard rod gas

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.