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arXiv:2604.21553

Dean-Kawasaki fluctuating hydrodynamics for backscattering hard rods

Mrinal Jyoti Powdel

Published Apr 23, 2026• 0 citations
arXivHTMLPDF

Abstract

We study a system of backscattering hard rods in one dimension. Contrary to the usual ballistic hard rods, these hard rods flip the sign of their velocities with a rate $γ$. This leads to the decay of the odd moments of velocity while preserving the even moments: the number of conserved quantities in the system becomes half. The introduction of the flipping rate $γ$ is an integrability-breaking perturbation, and this leads to a change in the transport properties in the system. We show using a Dean-Kawasaki fluctuating hydrodynamic formulation that the unequal space-time correlation of the normal mode phase space densities attains a diffusive form at late times. Also, we show that for $t \gg 1/γ$, the two-time density-density correlation of mass densities spreads in a diffusive manner, and for $t \ll 1/γ$, the correlation spreads ballistically, for a background state given by the Boltzmann distribution. Our results present an elegant framework to study systems where integrability is broken by a stochastic noise.

DOI
10.48550/arxiv.2604.21553

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We study a system of backscattering hard rods in one dimension. Contrary to the usual ballistic hard rods, these hard rods flip the sign of their velocities with a rate γγγ. This leads to the decay of the odd moments of velocity while preserving the even moments: the number of conserved quantities in the system becomes half. The introduction of the flipping rate γγγ is an integrability-breaking perturbation, and this leads to a change in the transport properties in the system. We show using a Dean-Kawasaki fluctuating hydrodynamic formulation that the unequal space-time correlation of the normal mode phase space densities attains a diffusive form at late times. Also, we show that for t≫1/γt \gg 1/γt≫1/γ, the two-time density-density correlation of mass densities spreads in a diffusive manner, and for t≪1/γt \ll 1/γt≪1/γ, the correlation spreads ballistically, for a background state given by the Boltzmann distribution. Our results present an elegant framework to study systems where integrability is broken by a stochastic noise.
We analyze the onset of diffusive hydrodynamics in the one-dimensional hard-rod gas subject to stochastic backscattering. While this perturbation breaks integrability and leads to a crossover from ballistic to diffusive transport, it preserves infinitely many conserved quantities corresponding to even moments of the velocity distribution of the gas. In the limit of small noise, we derive the exact expressions for the diffusion and structure factor matrices, and show that they generically have off diagonal components. We find that the particle density structure factor is non-Gaussian and singular near the origin, with a return probability showing logarithmic deviations from diffusion.
We consider a one-dimensional gas of hard rods, one of the simplest examples of an interacting integrable model. It is well known that the hydrodynamics of such integrable models can be understood by viewing the system as a gas of quasiparticles. Here, we explore the dynamics of individual quasiparticles for a variety of initial conditions of the background gas. The mean, variance, and two-time correlations are computed exactly and lead to a picture of quasiparticles as drifting Brownian particles. For the case of a homogeneous background, we show that the motion of two tagged quasiparticles is strongly correlated, and they move like a rigid rod at late times. Apart from a microscopic derivation based on the mapping to point particles, we provide an alternate derivation which emphasizes that quasiparticle fluctuations are related to initial phase-space fluctuations, which are carried over in time by Euler scale dynamics. For the homogeneous state, we use the Brownian motion picture to develop a Dean-Kawasaki-type fluctuating hydrodynamic theory, formally having the same structure as that derived recently by Ferrari and Olla. We discuss differences with existing proposals on the hydrodynamics of hard rods and some puzzles.
Abstract Recently, a theoretical framework known as 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 characterized 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.
Abstract We consider a system of many hard rods moving in one dimension. As it is an integrable system, it possesses an extensive number of conserved quantities and its evolution on a macroscopic scale can be described by generalized hydrodynamics. Using a microscopic approach, we compute the evolution of the conserved densities starting from non-equilibrium initial conditions of both quenched and annealed type. In addition to being reduced to the Euler solutions of the hydrodynamics in the thermodynamic limit, the microscopic solutions can also capture the effects of the Navier–Stokes terms and thus go beyond the Euler solutions. We demonstrate this feature from microscopic analysis and from numerical solution of the Navier–Stokes equation in two problems—first, tracer diffusion in a background of hard rods and, second, the evolution from a domain wall initial condition in which the velocity distribution of the rods is different on the two sides of the interface. We supplement our analytical results with extensive numerical simulations.
Abstract We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier–Stokes (NS) equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments.
We describe the crossover from generalized to conventional hydrodynamics in nearly integrable systems. Integrable systems have infinitely many conserved quantities, which spread ballistically, in general. When integrability is broken, only a few of these conserved quantities survive. The remaining conserved quantities are generically transported diffusively; we derive a compact and general diffusion equation for these. The diffusion constant depends on the matrix elements of the integrability-breaking perturbation; for a certain class of integrability-breaking perturbations, including long-range interactions, the diffusion constant can be expressed entirely in terms of generalized hydrodynamic data.