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arXiv:2309.14476 · Physical Review Letters

Navier-Stokes Equations for Low-Temperature One-Dimensional Quantum Fluids

Andrew Urichuk, Stefano Scopa, Jacopo De Nardis

Published Jun 11, 2024• 11 citations
arXivHTMLPDF

Abstract

We consider one-dimensional interacting quantum fluids, such as the Lieb-Liniger gas. By computing the low-temperature limit of its (generalized) hydrodynamics we show how in this limit the gas is well described by a conventional viscous (Navier-Stokes) hydrodynamics for density, fluid velocity, and the local temperature, and the other generalized temperatures in the case of integrable gases. The dynamic viscosity is proportional to temperature and can be expressed in a universal form only in terms of the emergent Luttinger liquid parameter K and its density. We show that the heating factor is finite even in the zero temperature limit, which implies that viscous contribution remains relevant also at zero temperatures. Moreover, we find that in the semiclassical limit of small couplings, kinematic viscosity diverges, reconciling with previous observations of Kardar-Parisi-Zhang fluctuations in mean-field quantum fluids.

DOI
10.1103/physrevlett.132.243402

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We consider one-dimensional interacting quantum fluids, such as the Lieb-Liniger gas. By computing the low-temperature limit of its (generalized) hydrodynamics we show how in this limit the gas is well described by a conventional viscous (Navier-Stokes) hydrodynamics for density, fluid velocity, and the local temperature, and the other generalized temperatures in the case of integrable gases. The dynamic viscosity is proportional to temperature and can be expressed in a universal form only in terms of the emergent Luttinger liquid parameter K and its density. We show that the heating factor is finite even in the zero temperature limit, which implies that viscous contribution remains relevant also at zero temperatures. Moreover, we find that in the semiclassical limit of small couplings, kinematic viscosity diverges, reconciling with previous observations of Kardar-Parisi-Zhang fluctuations in mean-field quantum fluids.
The Navier-Stokes equations are paradigmatic equations describing hydrodynamics of an interacting system with microscopic interactions encoded in transport coefficients. In this work we show how the Navier-Stokes equations arise from the microscopic dynamics of nearly integrable 1D quantum many-body systems. We build upon the recently developed hydrodynamics of integrable models to study the effective Boltzmann equation with collision integral taking into account the non-integrable interactions. We compute the transport coefficients and find that the resulting Navier-Stokes equations have two regimes, which differ in the viscous properties of the fluid. We illustrate the method by computing the transport coefficients for an experimentally relevant case of coupled 1D cold-atomic gases.
We develop a theory of viscous dissipation in one-dimensional single-component quantum liquids at low temperatures. Such liquids are characterized by a single viscosity coefficient, the bulk viscosity. We show that for a generic interaction between the constituent particles this viscosity diverges in the zero-temperature limit. In the special case of integrable models, the viscosity is infinite at any temperature, which can be interpreted as a breakdown of the hydrodynamic description. Our consideration is applicable to all single-component Galilean-invariant one-dimensional quantum liquids, regardless of the statistics of the constituent particles and the interaction strength.
The transport properties of one-dimensional Fermi gases at low temperatures are often described by the Luttinger liquid (LL) model. However, to study dissipative effects, one needs to examine interactions beyond the LL model. In this work, we provide a simple model that allows for a direct microscopic calculation of the bulk viscosity, namely, the one-dimensional spin polarized ppp-wave Fermi gas. To leading order in the finite interaction strength, we find that the bulk viscosity is finite and consistent with the requirement of scale symmetry. We further show that the bulk viscosity satisfies the Bose-Fermi duality relating the weakly interacting limit of the spin polarized Fermi gas to the strongly interacting limit of the Lieb-Liniger model and vice versa. This work establishes the bulk viscosity to leading order in scale-breaking interactions in both the strongly and weakly interacting limits for both high and low temperatures.
Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.
We develop the theory of hydrodynamic fluctuations for one-dimensional quantum liquids. At low temperatures the relaxation of interacting electrons in a single-channel wire separates into two classes: equilibration, which changes the numbers of right and left movers through backscattering and is exponentially slow, and thermalization, which redistributes energy and momentum at a fixed number of movers at a power-law rate. This separation of time scales gives the electron liquid a two-fluid description analogous to superfluid 4^44He, with the zero mode of the Luttinger liquid as the superfluid component and the gas of thermal excitations as the normal component. Using Onsager reciprocity and the fluctuation-dissipation theorem, we add Langevin fluxes to the two-fluid hydrodynamic equations, including a mutual-friction noise generated by rare backscattering events that has no counterpart in helium. We apply the theory to Coulomb-coupled quantum wires and calculate the dynamic structure factors, which contain a second-sound peak next to the plasmon doublet; the Coulomb drag resistivity, where intrawire equilibration acts as a bottleneck and turns the hydrodynamic drag into an activated temperature dependence; and the near-field heat transfer between the wires. Mode frequencies, spectral weights, linewidths, and the crossover scales of the transport coefficients are expressed through the thermodynamic properties of the liquid and a few dissipative coefficients, providing spectroscopic and transport probes of two-fluid behavior in one-dimensional quantum systems.
The aim of this paper is to derive the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker--Planck type diffusion of the probability density, called of Doebner--Goldin. This 'Doebner--Goldin hydrodynamic system' is shown to be reduced in most cases to a simpler one of quantum Euler type by means of the introduction of a nonlinear gauge transformation that changes the fluid mean velocity into a new effective velocity corrected by an osmotic contribution. Finally, we also discuss some particular situations of especial interest and compare the structure of the resulting fluid systems with that of the viscous quantum hydrodynamic and the quantum Navier--Stokes equations stemming from maximization of the quantum entropy for Wigner--BGK models.