Simulating generalised fluids via interacting wave packets evolution

TL;DR

This work addresses the challenge of simulating generalized hydrodynamics (GHD) for 1D integrable and near-integrable systems, including fluctuations and integrability-breaking perturbations. It introduces the Wave Packet Gas (WPG), a semiclassical particle-based representation that maps interacting quasiparticle dynamics to bare-particle trajectories, yielding an efficient, all-orders numerical framework that recovers GHD in integrable limits while automatically incorporating fluctuations and two-point correlations. The authors develop both classical (hard-rod) and generic (quantum-statistical) WPG mappings, derive how external potentials and two-body interactions modify the bare-particle dynamics, and demonstrate through several scenarios (cosine traps, harmonic/quartic traps, and dipolar-like interactions) that one-point observables may thermalize while long-range two-point correlations persist, challenging naive thermalization narratives. The approach enables fast, large-scale simulations of quasi-integrable systems and provides new insight into relaxation dynamics and correlation structure in 1D fluids, with direct relevance to cold-atom experiments and beyond.

Abstract

One-dimensional integrable and quasi-integrable systems display, on macroscopic scales, a universal form of transport known as Generalized Hydrodynamics (GHD). In its standard Euler-scale formulation, GHD mirrors the equations of a two-dimensional compressible fluid but ignores fluctuations and becomes numerically unwieldy as soon as integrability-breaking perturbations are introduced. We show that GHD can be efficiently simulated as a gas of semiclassical wave packets - a natural generalisation of hard-rod particles - whose trajectories are efficiently mapped onto those of point particles. This representation (i) provides a transparent route to incorporate integrability-breaking terms, and (ii) automatically embeds the exact fluctuating-hydrodynamics extension of GHD. The resulting framework enables fast, large-scale simulations of quasi-integrable systems even in the presence of complicated integrability-breaking perturbations. It also manifest the pivotal role of two-point correlations in systems confined by external potentials: we demonstrate that situations where local one-point observables appear thermalised can nevertheless sustain long-lived, far-from-equilibrium long-range correlations for arbitrarily long times, signaling that, differently from what previously stated, true thermalisation is not reached at diffusive time-scales.

Figures (8)

• Figure 1: Example trajectories of the WPG gas, with coordinates $x$, as obtained by mapping from the bare coordinates $X$. By choosing an appropriate scattering shift $\mathfrak{a}$ the interacting coordinates are realized for different models: pictured are the repulsive hard rod gas $\mathfrak{a}(\theta,\theta') = -1$ (upper left), attractive hard rod gas with $\mathfrak{a}(\theta,\theta') = 1$ (center), and Lieb-Liniger WPG with $\mathfrak{a}(\theta,\theta') = 2c/((\theta-\theta')^2 + c^2)$ and $c=1$ (upper right), all quenched from a homogeneous initial state to a cosine potential with the interacting dynamics determined from the bare coordinates (bottom). Notice how the attractive hard rod gas has similar scattering dynamics as the Lieb-Liniger gas and therefore can be used as a simpler theoretical toy model.
• Figure 2: Evolution of a homogeneous WPG sampled with Fermi-Dirac statistics using the scattering kernel of the Lieb-Liniger model with $\beta = c = \mu = 1$ and with $dX = 40$. A tracing particle is placed in the center of the system with $\theta = -2,-1,0,1,2$ and the displacement $\Delta x = x(t) - x(0)$ of this particle from its initial position is considered (left) the average displacement $\langle x \rangle$ approaches the corresponding effective velocity from GHD (right) the second moment of displacement approaches the diffusive constant predicted from Eq. \ref{['eq:diffuspreding']}.
• Figure 3: The two columns show respectively the dynamics of classical WPG with $\mathfrak{a}_{ij} = -1$ (left, repulsive) and $\mathfrak{a}_{ij} = +1$ (right, attractive). The initial state is a Bragg pulse, given by $\rho = \sum_{s=\pm} \exp{-(\theta -\theta_{s})^2/(2 T_0)}$ with $\theta_{\pm}=\pm 1$. The system is quenched into a cosine potential $V(x) = V_0 \cos( 2\pi x/\ell)$ with $\ell = 10$ and $V_0=1.0$. The top plots show the dynamics of kurtosis of the rapidity distribution as function of rescaled time $t/\ell$ for $T_0=1.0$ and $T_0=0.1$, where solid lines represent the WPG numerics and dashed lines the solution of Navier-Stokes GHD \ref{['eq:nabla_eq']}. The insets show the dynamics of rescaled rapidity distribution for $T_0=0.01$, where different colors represent different times, according to the colorbars shown in the plots below. The solid black line shows a featureless Gaussian distribution, while the dashed line represent the initial distribution. The second and third lines show respectively the dynamics of density and energy correlations, defined in \ref{['eq: def_sp_en_corr']}. Different colors represent different increasing times from light to dark, according to the colorbars.
• Figure 4: The four columns show respectively the dynamics of repulsive hard rods evolved in harmonic potential (I), repulsive hard rods evolved in quartic potential (II), attractive hard rods evolved in harmonic potential (III) and attractive hard rods evolved in quartic potential (IV). In all the case we set hard rods length $|\mathfrak{a}| = 1$ and we use $\omega_0 = \sqrt{50}$ for the repulsive case and for the attractive quartic case, while $\omega_0 = \sqrt{5}$ for the attractive harmonic. The initial state is homogeneous in the interval $[-\ell/2,\ell/2]$ with an average number of particles $N=\ell/2=50$ and is prepared in a Bragg pulse, given by rapidity distribution $\rho = \sum_{s=\pm} \exp{-(\theta -\theta_{s})^2/(2 T_0)}$, with $\theta = \pm 1$ and $T_0=0.1$. The first line shows the WPG dynamics of kurtosis of rapidity distribution against rescaled time $t/\ell$. The insets show the dynamics of rescaled rapidity distribution, where different colors represent different time data according to the colorbars shown in the plots below. The solid black line shows a featureless Gaussian distribution, while the dashed line represent the initial distribution. The second and third lines respectively show the dynamics of space and energy correlations defined in \ref{['eq: def_sp_en_corr']}, computed at $y=0$. Different colors represent different increasing times from light to dark, according to the colorbars. Dashed and solid lines are respectively the initial and final time data.
• Figure 5: Same as in Fig. \ref{['fig:HarmonicQuartic-1']} with the only difference in the initial state preparation: we initialize the system in a thermal state in the interval $[-\ell,\ell]$ with temperature $T_0=0.01$ and $N=\ell=100$, i.e. homogeneous in space and with Gaussian rapidity distribution. Then, at time $t=0$, the particles in $[-\ell/2,\ell/2]$ are removed and we evolve the two chopped clouds.