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arXiv:2406.18322 · Physical review. B./Physical review. B

Quadrature for the generalized hydrodynamics equation and absence of shocks in the Lieb-Liniger model

Friedrich Hübner, Benjamin Doyon

Published Sep 26, 2025• 2 citations
arXivHTMLPDF

Abstract

In conventional fluids, it is well known that Euler-scale equations are plagued by ambiguiti es and instabilities. In one dimension smooth initial conditions may develop shocks, and weak solutions, such as for domain wall initial conditions (the paradigmatic Riemann problem of hydrodynamics), are not unique (and must be supplemented by physically motivated conditions such as non-negative entropy production). The absence of shock formation experimentally observed in quasi-one-dimensional cold-atomic gases, which are described by the Lieb-Liniger model, provides perhaps the strongest pointer to a modification of the hydrodynamic equation due to integrability. Generalized hydrodynamics (GHD) is the required hydrodynamic theory, taking into account the infinite number of conserved quantities afforded by integrability. We provide a new quadrature for the GHD equation—a solution in terms of a Banach fixed-point problem where time has been explicitly integrated. The quadrature is an efficient numerical solution tool; and it allows us, in the Lieb-Liniger model, to rigorously show that no shock may appear at all times, and, when combined with recent hydrodynamic fluctuation theories, to obtain new expressions for correlations in nonstationary states, establishing for the first time the presence of discontinuities that are fundamental to the nonequilibrium physics of integrable systems.

DOI
10.1103/5w4p-bxnt

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In conventional fluids, it is well known that Euler-scale equations are plagued by ambiguiti es and instabilities. In one dimension smooth initial conditions may develop shocks, and weak solutions, such as for domain wall initial conditions (the paradigmatic Riemann problem of hydrodynamics), are not unique (and must be supplemented by physically motivated conditions such as non-negative entropy production). The absence of shock formation experimentally observed in quasi-one-dimensional cold-atomic gases, which are described by the Lieb-Liniger model, provides perhaps the strongest pointer to a modification of the hydrodynamic equation due to integrability. Generalized hydrodynamics (GHD) is the required hydrodynamic theory, taking into account the infinite number of conserved quantities afforded by integrability. We provide a new quadrature for the GHD equation—a solution in terms of a Banach fixed-point problem where time has been explicitly integrated. The quadrature is an efficient numerical solution tool; and it allows us, in the Lieb-Liniger model, to rigorously show that no shock may appear at all times, and, when combined with recent hydrodynamic fluctuation theories, to obtain new expressions for correlations in nonstationary states, establishing for the first time the presence of discontinuities that are fundamental to the nonequilibrium physics of integrable systems.
The generalized hydrodynamics (GHD) equation is the equivalent of the Euler equations of hydrodynamics for integrable models. Systems of hyperbolic equations such as the Euler equations usually develop shocks and are plagued by problems of uniqueness. We establish for the first time the existence and uniqueness of solutions to the full GHD equation and the absence of shocks, from a large class of initial conditions with bounded occupation function. We assume only absolute integrability of the two-body scattering shift. In applications to quantum models of fermionic type, this includes all commonly used physical initial states, such as locally thermal states and zero-entropy states. We show in particular that differentiable initial conditions give differentiable solutions at all times and that weak initial conditions such as the Riemann problem have unique weak solutions which preserve entropy. For this purpose, we write the GHD equation as a new fixed-point problem (announced in a companion paper). We show that the fixed point exists, is unique, and is approached, under an iterative solution procedure, in the Banach topology on functions of momenta.
The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. In this Letter, we show that it supersedes the widely used conventional hydrodynamics (CHD) of one-dimensional Bose gases. We illustrate this by studying "nonlinear sound waves" emanating from initial density accumulations in the Lieb-Liniger model. We show that, at zero temperature and in the absence of shocks, GHD reduces to CHD, thus for the first time justifying its use from purely hydrodynamic principles. We show that sharp profiles, which appear in finite times in CHD, immediately dissolve into a higher hierarchy of reductions of GHD, with no sustained shock. CHD thereon fails to capture the correct hydrodynamics. We establish the correct hydrodynamic equations, which are finite-dimensional reductions of GHD characterized by multiple, disjoint Fermi seas. We further verify that at nonzero temperature, CHD fails at all nonzero times. Finally, we numerically confirm the emergence of hydrodynamics at zero temperature by comparing its predictions with a full quantum simulation performed using the NRG-TSA-abacus algorithm. The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.
Generalized hydrodynamics is a framework to study the large scale dynamics of integrable models, special fine-tuned one-dimensional many-body systems that possess an infinite number of local conserved quantities. Unlike classical models, where the microscopic origins of generalized hydrodynamics are better understood, in quantum models it can only be derived using the hydrodynamic formalism. Using the paradigmatic and experimentally relevant repulsive Lieb-Liniger model as an example, this thesis introduces a new viewpoint on the dynamics of quantum integrable models by introducing so-called semi-classical Bethe models. These classical integrable models act as an intermediate description between the microscopic quantum realm and the macroscopic generalized hydrodynamics. After introducing these models and discussing their properties, we study the generalized hydrodynamics equation using new tools and show that solutions to the Euler generalized hydrodynamics equation of the Lieb-Liniger model exist, are unique and do not develop gradient catastrophes. Finally, we discuss new insights into the physics governing the diffusive correction, which, contrary to prior belief, is not described by a Navier-Stokes-like equation. Focusing on the main intuitive ideas, the thesis aims to provide a self-contained overview over these exciting new developments on generalized hydrodynamics.
Generalized hydrodynamics (GHD) was proposed recently as a formulation of hydrodynamics for integrable systems, taking into account infinitely-many conservation laws. In this note we further develop the theory in various directions. By extending GHD to all commuting flows of the integrable model, we provide a full description of how to take into account weakly varying force fields, temperature fields and other inhomogeneous external fields within GHD. We expect this can be used, for instance, to characterize the non-equilibrium dynamics of one-dimensional Bose gases in trap potentials. We further show how the equations of state at the core of GHD follow from the continuity relation for entropy, and we show how to recover Euler-like equations and discuss possible viscosity terms.
The results of benchmarks of the recent theory of generalized hydrodynamics (GHD) in the context of the dynamics of the 1D Bose gas described by the Lieb-Liniger model are reported. Through comparisons against several alternative, well-established computational approaches, the range of applicability of GHD in some of the most challenging out-of-equilibrium scenarios, such as propagation of quantum shock waves and collisional relaxation, is verified and validated and, additionally, some of the methods limitations are pointed out.
We show that the equations of generalized hydrodynamics (GHD), a hydrodynamic theory for integrable quantum systems at the Euler scale, emerge in full generality in a family of classical gases, which generalize the gas of hard rods. In this family, the particles, upon colliding, jump forward or backward by a distance that depends on their velocities, reminiscent of classical soliton scattering. This provides a "molecular dynamics" for GHD: a numerical solver which is efficient, flexible, and which applies to the presence of external force fields. GHD also describes the hydrodynamics of classical soliton gases. We identify the GHD of any quantum model with that of the gas of its solitonlike wave packets, thus providing a remarkable quantum-classical equivalence. The theory is directly applicable, for instance, to integrable quantum chains and to the Lieb-Liniger model realized in cold-atom experiments.