Microscopic and hydrodynamic correlation in 1d hard rod gas

TL;DR

This work addresses the problem of understanding space-time correlations in a one-dimensional gas of hard rods by performing exact microscopic calculations of mass-density correlations and comparing them to macroscopic predictions from ballistic macroscopic fluctuation theory (BMFT). The authors map hard rods to hard-point particles to obtain microscopic correlators and then implement a systematic coarse-graining to define a coarse-grained density, whose fluctuations are described by BMFT on the Euler (ballistic) scale. They demonstrate quantitative agreement between microscopically computed correlations and BMFT predictions, thereby validating the micro–macro correspondence and the core BMFT assumptions. The results establish long-range Euler-scale correlations in this integrable system and pave the way for applying BMFT to other observables and initial conditions in 1D interacting systems. The work has implications for how hydrodynamic theories connect to microscopic dynamics in integrable models and provides a concrete benchmark for the BMFT framework.

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.

Figures (5)

• Figure 1: Comparison of the density profiles of the special component obtained from the expression in Eq. \ref{['sol:beyond-Euler-2']}, and numerical simulations at $t=40,$ for $\hat{\varrho}_0=0.5.$ Here, the blue dashed line denotes simulation data, red circles represent the analytical solution [Eq. \ref{['sol:beyond-Euler-2']}], and green solid line corresponds to the Euler GHD solution Mrinal_2024_HRsingh2024thermalization The profile is shown near the shock region, where the large $N$ analytical result exhibits excellent agreement with the numerical data, compared to the Euler GHD solution. The other parameters are $N=1000, a=0.01, v_0=1$, and averages are taken over $10^5$ realizations.
• Figure 2: The plot (a) shows the evolution of the two-time density correlation $\mathscr{C}(X,t;Y,0)$ at $t=1,5,10$ for fixed $Y=48.91$ using the analytical exact expressions following Eq. \ref{['eq:conn_corr_two_time']}. The plot (b) shows the comparison of the numerical simulation (blue circles) with the analytical exact result (red line) for $t=10$. using the mapping of Eq. \ref{['eq:HR to HP mapping infinite line']}. The parameters considered here are $N=200,~a=0.02,~\sigma=20$, and $T=1$. For numerical results, the averages have been taken over $5 \times 10^{10}$ initial configurations.
• Figure 3: Plots comparing the mass density correlation $\mathscr{C}(X,t;Y,t)$ between the analytical exact expression (red solid lines) following Eq. \ref{['eq:jont_dist_same_t_two_point']} and the numerical simulation (blue circles) for $Y=1.55$ and $t=5$ at different levels of magnification. Plot (a) shows the correlation over the full spatial range in $X$. Plot (b) presents a zoomed-in view around $X=Y$ position. To further resolve the local structure of the correlation around $X=Y$, plot (c) displays a finer zoom with more closely spaced data points. The values of other parameters considered for this plot are $N=500,a=0.1, \sigma=0.5,$ and $T=1$ and $Y=1.55.$ The averages have been taken over $10^7$ initial configurations.
• Figure 4: Plots compairing the equal-time connected correlation function derived from BMFT as in Eq. \ref{['eq:corr_expr_bmft']} [blue solid line], with that obtained by coarse-graining the analytical microscopic expression $\mathscr{C}(X,t;Y,t)$ (given in Eq. \ref{["eq:rxt_ryt'_defn"]} using Eqs. \ref{['eq:rho_Xt_micro']} and \ref{['eq:rxryt_expr_full']}) [orange discs], and the numerical simulation of the microscopic dynamics [blue star], for $N=200, a=0.002$. The coarse-graining has been performed for the macroscopic position $\mathtt{X}_b=0.5135$ and $\mathtt{t}=0.5$ using fluid-cell averaging as described in Eq. \ref{['def:rho_cg']}, with the fluid cell size chosen as $\Delta X=0.1\ell,$ where $\ell=20.$ The coarse-grained equal-time correlation obtained from the microscopic dynamics shows very good agreement with the long-range correlations obtained theoretically by BMFT. Additionally, we plot the coarse-grained correlation obtained from microscopic correlation for $N=400,600$ [triangle and square symbols] using the identical initial condition with $\ell=40$ and $60$ , respectively, for $a=0.002$, and at the same fixed macroscopic time $\mathtt{t}=0.5.$ The excellent collapse of the coarse-grained correlation functions for different $N$ and $\ell$ (with same ratio $\frac{\ell}{Na}$) verifies the scaling relation in Eq. \ref{['eq:corr_scaling']}. Inset: Plots of the corresponding microscopic correlation $\mathscr{C}(X,t;Y,t)$ for $N=200$ as functions of $X$ for few values of $Y$ fixed in the range $[ \mathtt{X}_b \ell- 0.1\ell, \mathtt{X}_b \ell+ 0.1\ell]$ with $\mathtt{X}_b=0.5135$. These microscopic correlation functions are used on the right hand side of Eq. \ref{['rel:corr-macro-micro']} at the coarse-graining step to obtain the hydro-scale correlation plotted in the main figure.
• Figure 5: Comparison of the two-time connected correlation function obtained from BMFT (blue solid line) as in Eq. \ref{['eq:twotime_bmft_full']}, with that derived by coarse-graining the analytical microscopic expression in Eqs. \ref{['eq:conn_corr_two_time']} (orange discs) and the numerical simulation of the microscopic dynamics (stars). The coarse-graining has been performed for the macroscopic time $\mathtt{t}=0.2$ at the macroscopic position $\mathtt{X}_b=2.4455$ using fluid-cell averaging as described in Eq. \ref{['def:rho_cg']}, with the fluid cell size chosen as $\Delta X=0.1\ell,$ where $\ell=20.$ The coarse-grained equal-time correlation obtained from the microscopic dynamics shows an excellent agreement with the same obtained theoretically by BMFT. For this plot, we choose $N=200$ and $a=0.02$. The averages have been taken over $10^9$ initial configurations.