Quasiparticle dynamics and hydrodynamics of 1d hard rod gas on diffusion scale

We investigate the stochastic dynamics of a quasiparticle within a gas of hard rods, focusing on the evolution of its mean, variance, and autocorrelation for two choices of initial states: (i) one with long-range (LR) correlations and (ii) the other without it. We derive analytical results for the phase space density correlations in the former case to complement the known results for the latter case. These results enable us to obtain expressions for the mean, variance, and autocorrelation of a quasiparticle, which are applicable to both initial states. The LR correlations introduce a diffusive-scale correction to the mean Euler generalized hydrodynamic (GHD) equations, modifying the standard local equilibrium form, and our findings reveal that the form of the correction term depends on the LR correlations present in the initial state.

💡 Research Summary

This paper investigates the stochastic dynamics of a quasiparticle (often called a “tagged rod”) moving in a one‑dimensional gas of hard rods, with a particular focus on how the quasiparticle’s mean position, variance, and autocorrelation evolve over time. Two families of initial conditions are considered: (i) a state with long‑range (LR) correlations already present in the microscopic configuration (IC fhp), and (ii) a state without such correlations (IC fhr). The authors first map the hard‑rod system onto an equivalent point‑particle system via the coordinate transformation (x_i = X_i-(i-1)a). This mapping removes the finite rod length from the dynamics, leaving a set of non‑interacting point particles that exchange velocities upon collision; the original rod dynamics are recovered by re‑ordering the particles after each collision.

In the point‑particle picture the initial distribution can be factorized either directly in rod coordinates (IC fhr, Eq. (3)) or in point‑particle coordinates (IC fhp, Eq. (5)). The latter already encodes LR correlations because the ordering constraint (\Theta(x_{i+1}-x_i)) couples distant particles through the density profile. Both ensembles are taken in the thermodynamic limit (N\to\infty), (L\to\infty) with a smooth macroscopic density (\rho_0(X)).

The coarse‑grained phase‑space density (\hat f(X,v,t)) is defined by averaging the microscopic empirical density over a mesoscopic window (\Delta X) that satisfies (a\ll\Delta X\ll\ell), where (\ell) is the scale over which the initial profile varies. In the Euler (ballistic) scaling limit ((X\sim t)) the average density (\bar f) obeys the standard generalized hydrodynamics (GHD) equation \