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arXiv:2510.19693 · SciPost Physics

Stochastic dynamics of quasiparticles in the hard rod gas

Seema Chahal, Indranil Mukherjee, Abhishek Dhar, Herbert E. Spohn, Anupam Kundu

Published Jun 16, 2026• 0 citations
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Abstract

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.

DOI
10.21468/scipostphys.20.6.166

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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 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 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 devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.
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 Inspired by recent results on the non-equilibrium dynamics of many-body quantum systems, we study the classical hard rod problem in one dimension with initial domain wall condition. Hard rods are an integrable system, in the sense that for each velocity the density of particles is locally conserved. It was proven by Boldrighini et al (1983 J. Stat. Phys . 31 577) that on the hydrodynamic space-time scale, the fluid of hard rods satisfies Euler-type equations which comprise all conservation laws. We provide the general solution to these equations on the line, with an initial condition where the left and right halves are, asymptotically, in different states. The solution is interpreted as being composed of a continuum of contact discontinuities, one for each velocity. This is a classical counterpart of the transport problem solved recently in quantum integrable systems. We also discuss the Navier–Stokes (viscous) corrections, and study its effect on the broadening of the contact discontinuity and on entropy production.