Diffusive hydrodynamics of hard rods from microscopics

TL;DR

This work derives exact, microscopically grounded diffusive hydrodynamics for a one-dimensional hard-rod gas, showing that ballistic (Euler) dynamics are supplemented by diffusion arising from long-range correlations. The authors develop two coupled equations: one for the one-point quasi-particle density $ ho(x,p)$ and another for the connected two-point correlation $C$, capturing how LR correlations influence diffusive transport. A key finding is that a cancellation occurs between singular local-GGE contributions and a jump in LR correlations at $x=y$, leaving a diffusion term governed by the symmetric LR part and yielding time-reversal invariant dynamics, rather than the entropy-increasing Navier–Stokes form. When starting from local equilibrium, LR correlations are absent and the diffusion reduces to the traditional NS-like form; under evolution, LR correlations emerge and dominate the diffusive scale, connecting microscopic ballistic dynamics to macroscopic diffusion and long-range fluctuations. The results provide a first exact microscopic realization of how ballistic dynamics generate long-range correlations that alter diffusive hydrodynamics, aligning with ballistic macroscopic fluctuation theory and offering a concrete framework for further exploration of integrable and near-integrable systems.

Abstract

We derive exact equations governing the large-scale dynamics of hard rods, including diffusive effects that go beyond ballistic transport. Diffusive corrections are the first-order terms in the hydrodynamic gradient expansion and we obtain them through an explicit microscopic calculation of the dynamics of hard rods. We show that they differ significantly from the prediction of Navier-Stokes hydrodynamics, as the correct hydrodynamics description is instead given by two coupled equations, giving respectively the evolution of the one point functions and of the connected two-point correlations. The resulting equations are time-reversible and reduce to the usual Navier-Stokes hydrodynamic equations in the limit of near-equilibrium evolution. This represents the first exact microscopic calculation showing how ballistic dynamics generates long-range correlations, in agreement with general results from the recently developed ballistic macroscopic fluctuation theory, and showing how such long range-correlations directly affect the diffusive hydrodynamic terms, in agreement with, and clarifying, recent related results.

Figures (6)

• Figure 1: Explanation of long range correlations: As time evolves (see \ref{['equ:overview_basic_evolution']}) two hard rods (in this case the blue and the orange one), become correlated because they interact with the same particles between them. If the number of particles between them is lower, then they will travel less far, if they are higher, both will travel further. Since the density of particles in a region fluctuates around its mean value, this means that particles which travel through the same region become correlated. In the hydrodynamic limit, this can then be observed as non-trivial long-range correlations of the density.
• Figure 2: Integrated correlations $\int\dd{p}\dd{q} q C(x,y,p,q,t)$ at $y=0.7$ and $t=1$, as function of $x$. The initial state is defined in Eq. \ref{['eqn: initial state num']}. The black circles represent the numerical simulation (which are done as described in section \ref{['sec:numerics']}), while the red curve the theoretical prediction, see Appendix \ref{['app:correlation']}. This figure shows how the system developed long range correlations during the dynamics and that they are perfectly captured by the theory. Also, at $x=y$, we observe a discontinuity. For numerical simulation we use the same data as in Hubner2024DiffusionLongRange.
• Figure 3: Evolution of $n$-moments $\langle p^n\rangle(x,t)$ of hard rods velocity distribution from the initial state \ref{['eqn: initial state num']} as a function of space, for $n\in\{0,1,2,3\}$ and for $t\in\{0,0.5,1,1,5,2\}$. Empty circles represent the numerical simulation of the hard rods gas with $\ell=200$, while red lines represent the theoretical prediction of Eq. \ref{['equ:overview_basic_hatx']}. The numerical data are averaged over an ensemble of $3\times10^6$ different realizations.
• Figure 4: Time evolution of the $O(1/\ell)$ correction $\Delta\langle p^n\rangle_{1/\ell}$ to the $n$-moments of hard rods velocity distribution from the initial state \ref{['eqn: initial state num']} as a function of space, for $n\in\{0,1,2,3\}$ and for $t\in\{0,0.5,1,1,5,2\}$. The black line represent the numerical simulation of the hard rods gas, evaluated as the $f_2$ parameter of a fit with model $f(\ell)=f_1+f_2/\ell$ to $\langle p^n\rangle(x,t;\ell)$, for $\ell\in\{100,120,140,160,180,200\}$. The fit is performed independently for each point in space and time and the associated error bar is the standard deviation of the fit parameter. The numerical data are averaged over an ensemble of $3\times10^6$ different realizations. The red lines represent the theoretical prediction of Eq. \ref{['equ:expansion_integrated_test_funtion']}.
• Figure 5: We show the $O(1/\ell)$ correction to time derivative of the second moment of hard rods velocity distribution from the initial state \ref{['eqn: initial state num']}, as a function of space at $t=1$. The black line represent the numerical simulation of the hard rods gas, for which the time derivative is evaluated through the difference quotient \ref{['eqn: difference quotient']} with $\Delta t=0.05$. Then, the $O(1/\ell)$ correction is estimated as the $f_2$ parameter of a fit with model $f(\ell)=f_1+f_2/\ell$ to $\partial_t\langle p^2\rangle(x,t;\ell)$, for $\ell\in\{500,600,\ldots,1000\}$. The fit is performed independently for each point in space and the associated error bar is the standard deviation of the fit parameter. The numerical data are averaged over an ensemble of $8\times10^{10}$ different realizations. The red dots represent the theoretical prediction of the new theory Eq. \ref{['equ:diffusion_final_2']}, while green dots are the prediction due to the Navier-Stokes-like theory Eq. \ref{['equ:intro_NS']}. Finally, blue circles represent the time derivative of the solution \ref{['equ:rho_solution_explicit']}, computed through difference quotient with $\Delta t=0.01$. The inset shows the distance between theoretical predictions and data points in error bars units. One can clearly see that the new theory fits the numerical simulations much better, while the old Navier-Stokes-like theory Eq. \ref{['equ:intro_NS']} differs significantly. For numerical simulation we use the same data as in Hubner2024DiffusionLongRange.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4