Ballistic macroscopic fluctuation theory of correlations in hard rod gas

TL;DR

This work extends ballistic macroscopic fluctuation theory (BMFT) to a one-dimensional hard-rod gas, formulating fluctuations through a slowly varying phase-space density and deriving Euler-scale, two-point, two-time correlations of conserved densities. By mapping the hard-rod dynamics to a hard-point system, the authors obtain tractable saddle-point equations that yield explicit, numerically accessible expressions for the unequal space-time PSD correlations and, via hydrodynamic projections, for mass-density correlations. A central finding is the explicit demonstration of long-range correlations that develop under non-stationary, inhomogeneous initial states with interactions and at least two ballistic hydrodynamic modes, while equilibrium or single-mode limits suppress such correlations. The results connect BMFT with generalized hydrodynamics (GHD) concepts, reproduce known equilibrium expressions, and provide a practical route to compute fluctuations and correlations in integrable, ballistic systems with slow initial variations.

Abstract

Recently, a theoretical framework known as {\it ballistic macroscopic fluctuation theory} has been developed to study large-scale fluctuations and correlations in many-body systems exhibiting ballistic transport. In this paper, we review this theory in the context of a one-dimensional gas of hard rods. The initial configurations of the rods are sampled from a probability distribution characterised by slowly varying conserved density profiles across space. Beginning from a microscopic description, we first formulate the macroscopic fluctuation theory in terms of the phase-space density of quasiparticles. In the second part, we apply this framework to compute the two-point, two-time correlation functions of the conserved densities in the Euler scaling limit. We derive an explicit expression for the correlation function which not only reveals its inherent symmetries, but is also straightforward to evaluate numerically for a given initial state. Our results also recover known expressions for space-time correlations in equilibrium for the hard-rod gas.

Figures (2)

• Figure 1: Schematic diagram of a gas of $N$ hard rods, each of length $a$ and unit mass in one dimension. The space of the system is divided into blocks of size $\sim \ell$ and labelled by $\mathtt{y}_k=k\ell$. The number of rods also gets distributed among these blocks such that $\sum_kN_k=N$. We assume the following hierarchy of the length scales $a \ll \ell \ll Na$. We will eventuallyconsider the limit $N \to \infty$ and $\ell \to \infty$ however with $\ell/N \to 0$.
• Figure 2: Plot of the correlation $\mathscr{C}_{0,0}(x_a,t_a;x_b,t_b)$ at different times for the initial condition in Eq. \ref{['ini-psd']}. The plots (a), (b) and (c) correspond to the long-range part of the equal time correlations of mass densities and the plot in (d) corresponds to unequal space-time correlation for hard rods of length $a=1$. Recall, the densities at two distant locations at a later time $t$ get correlated (long-range correlation) because the fluctuations at these two locations originated from the same initial density fluctuation that coherently got carried to these locations by Euler evolution [see doyon2023ballistic for more physical insights]. Hence, for the initial condition in Eq. \ref{['ini-psd']}, long-range correlation can get developed only over a finite region over a given duration. The width of the region of course depends on $t_a,~t_b$ and the maximum relative speed of the particles, whereas the location of this region is decided by $x_b$. One finds that the ranges over which the long-range correlations get develop for the four plots are, (a) $[-2.002,2.002]$ (b) $[-1.016,1.016]$ (c) $[-0.218,3.359]$ and (d) $[0.342,2.91]$. These ranges can be identified from the $\Theta$ functions present in the explicit expression in Eqs. \ref{['eq:S_00(ta,tb)-fn']} and \ref{['eq:mcA']}. We observe discontinuities in the unequal space time correlation in plot (d) at locations $x_\pm=x_{\bar{\mathfrak{f}}(t_a)}(x'_a(\pm))$ with $x'_a(\pm)=x'_b \pm(t_a-t_b)$ for $x'_b=0.883~ (x_b=1.8),~t_a=0.2$ and $t_b=0.5$. It seems the discontinuities are artifacts of the discrete velocity distribution chosen in Eq. \ref{['ini-psd']}.