Burgers equation from non-thermal stationary states in nearly-integrable gases

TL;DR

The paper addresses diffusion in nearly-integrable, one-dimensional gases coupled weakly to a large bath. It combines generalized hydrodynamics with Chapman–Enskog theory to derive an effective hydrodynamic equation for the minority density, showing that a non-thermal, parity-breaking bath induces a Burgers-type equation with advection $u(\varrho)$ and diffusion $\mathcal{D}(\varrho)$; quantum statistics of the minority lead to nontrivial $u(\varrho)$, whereas classical cases reduce to diffusion after a Galilean boost. By analyzing Fermi’s Golden Rule collisions, a generalized BBGKY hierarchy, and minimal stochastic velocity-swap models, the authors compute transport coefficients and validate them against molecular-dynamics simulations of coupled Lieb–Liniger tubes and simplified collision models. Key contributions include explicit expressions for $\mathcal{D}(\varrho)$ and $u(\varrho)$, the demonstration of Burgers dynamics in a quantum setting, and a robust framework connecting gBBGKY, GHD, and kinetic theory for nearly integrable systems. The findings have significance for understanding transport in non-thermal environments and may inform experiments on quasi-1D cold-atom gases and related platforms where integrability breaking is weak but non-negligible.

Abstract

When a gas of particles interacts with much a larger reservoir the dynamics of density on large scales is typically governed by diffusion. We study this paradigmatic problem for weakly coupled integrable systems and show that this picture is altered when transport is investigated on top of long-lived non-thermal states. Remarkably, for states non-invariant under parity we find Burgers equation arising in the hydrodynamic limit. We explicitly compute the diffusion constant and nonlinear Euler-scale coupling of the Burgers equation using a variant of the Chapman-Enskog theory. We find excellent agreement between our theory and numerical simulations of a simplified model of stochastic two-body collisions, which we call velocity swap models. Our conclusions are based only on Galilean invariance, existence of a small system-bath coupling parameter and a small momentum exchange between the system and the bath.

Figures (6)

• Figure 1: We consider dynamics of particles with density $\varrho$ coupled to a much larger bath with density $\varrho^{(b)} \gg \varrho$. Both systems are described by integrable models and the long-range coupling breaks the integrability. One can think about two realizations of this scenario: mixed systems (a) and long-range interacting tubes (b). Particles of the observed system feature single conservation law and we consider the problem of hydrodynamics, which emerges on large space-time scales.
• Figure 2: Relaxation dynamics of kinetic energy per particle $e(t)$ in classical stochastic velocity swap model. The molecular simulation agrees very well with kinetic equation \ref{['eq:swap1']}. Inset: initial and final distributions of velocities. Solid lines are analytical distributions, in particular the final state agrees very well with \ref{['eq:Master_stat']}. We consider homogenous system of $N=1600$ particles with $\varrho=1, \varrho^{(b)}=100$ and with $\gamma=0.001$. The initial system state is thermal with temperature $T=0.08$, whereas the bath system state is asymmetric Bragg state $\rho_p^{(b)} \propto e^{-(\lambda+\lambda_B)^2/\sigma^2}+be^{-(\lambda-\lambda_B)^2/\sigma^2}$ with $\sigma=0.2$, $\lambda_B=0.4$ and $b=0.5$. The results were averaged over 1000 realizations.
• Figure 3: In the upper plot, comparison of diffusion coefficient stemming from RTA with non-approximated diffusion for the two-tubes system in a thermal state ($T = 4.0$) in Tonks--Girardeau limit. In the lower plot we show the function $\nu = u'/\mathcal{D}_{\rm{RTA}}$ for an asymmetric Bragg split. Blue color represents a system in a small temperature ($T = 1.0$), with the peak separation chosen in a way so that they overlap over a small region in $\lambda$. For the red-colored system we increased the temperature ($T = 10.0$), simultaneously increasing peak separation. By $\tau_{\rm{RTA,i}}$ we understand the characteristic relaxation time of the system in the blue/orange state
• Figure 4: Advective coefficient and diffusion coefficient as a function of $\varrho$ for the fermionic system coupled to the classical bath in the same state and $\omega$ as in Fig. \ref{['MolecularDynamicsPlots']}
• Figure 5: Molecular dynamics simulations for velocity swap models ($\gamma = 0.08$, $\varrho^{(b)} = 50.0$, $N=1600$) with initial states given by \ref{['eq:initial_states']}. We observe an excellent agreement between numerical simulations (histograms) and hydrodynamic equation \ref{['Burgers_intro']} derived through Chapman-Enskog method (solid lines) for all cases. For symmetric states in bath system, both systems are described by diffusion equation \ref{['eq:diff_eq']}. Upon introducing the asymmetry, the classical system is described by boosted diffusion equation. On the other hand, the fermionic model exhibits Burgers equation dynamics, as visible in the dynamically generated asymmetry in the initially Gaussian shape. Results from microscopic simulations are averaged over 1000 realizations.