The one-dimensional hard rod gas is one of the few exactly solvable many-body systems. Particles move freely between elastic collisions, and generalized hydrodynamics (GHD) provides an exact framework for the Euler-scale evolution — the leading-order behavior where the system is locally in equilibrium and gradients drive the dynamics.
Below Euler scale sits the diffusive scale, where fluctuations and correlations contribute corrections. The authors track a single quasiparticle in the hard rod gas and derive its mean, variance, and autocorrelation analytically. The surprising finding: when the initial state has long-range correlations, the diffusive correction to the GHD equations takes a different form than the standard one.
In the uncorrelated case, diffusive corrections are universal — they depend on local thermodynamic properties and have a standard structure derived from local equilibrium fluctuations. Long-range correlations break this universality. The correction term's structure depends on the specific form of the initial correlations, not just on local properties. Different initial states with the same local thermodynamics but different long-range structure produce different diffusive-scale behavior.
The structural point: the Euler scale is insensitive to initial correlations (it depends only on local equilibrium parameters), but the diffusive scale remembers them. The hierarchy of hydrodynamic scales is not just a matter of decreasing importance — it is a hierarchy of increasing sensitivity to initial conditions. What the leading order forgets, the next order retrieves.
(arXiv:2603.18522)