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arXiv:2410.15142 · Physical review. D/Physical review. D.

Relativistic second-order spin hydrodynamics: A Kubo-type formulation for the quark-gluon plasma

Duan She, Yi-Wei Qiu, Defu Hou

Published Feb 21, 2025• 10 citations
arXivHTMLPDF

Abstract

Building upon Zubarev's nonequilibrium statistical operator formalism, we derive a relativistic canonical-like second-order spin hydrodynamics for two power-counting schemes. We obtain comprehensive second-order expressions for dissipative fluxes, including the shear stress tensor, bulk viscous pressure, charge-diffusion currents, rotational stress tensor, boost heat vector, and spin tensor-related dissipative flux. By introducing novel transport coefficients and expressing them in terms of equilibrium correlation functions, we establish new Kubo-type formulas for second-order transport. Our findings have significant implications for understanding the collective spin dynamics of strongly interacting matter and provide a robust theoretical basis for future experimental and theoretical studies.

DOI
10.1103/physrevd.111.036027

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Building upon Zubarev's nonequilibrium statistical operator formalism, we derive a relativistic canonical-like second-order spin hydrodynamics for two power-counting schemes. We obtain comprehensive second-order expressions for dissipative fluxes, including the shear stress tensor, bulk viscous pressure, charge-diffusion currents, rotational stress tensor, boost heat vector, and spin tensor-related dissipative flux. By introducing novel transport coefficients and expressing them in terms of equilibrium correlation functions, we establish new Kubo-type formulas for second-order transport. Our findings have significant implications for understanding the collective spin dynamics of strongly interacting matter and provide a robust theoretical basis for future experimental and theoretical studies.
We derive Kubo formulae for first-order spin hydrodynamics based on nonequilibrium statistical operators method. In first-order spin hydrodynamics, there are two new transport coefficients besides the ordinary ones appearing in first-order viscous hydrodynamics. They emerge due to the incorporation of the spin degree of freedom into fluids and the spin-orbital coupling. Zubarev's nonequilibrium statistical operator method can be well applied to investigate these quantum effects in fluids. The Kubo formulae, based on the method of nonequilibrium statistical operators, are related to equilibrium (imaginary-time) infrared Green's functions, and all the transport coefficients can be determined when the microscopic theory is specified.
At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity $\ensuremath{\eta}$ and on five additional ``second-order'' hydrodynamical coefficients ${\ensuremath{\tau}}_{\ensuremath{\Pi}}$, $\ensuremath{\kappa}$, ${\ensuremath{\lambda}}_{1}$, ${\ensuremath{\lambda}}_{2}$, and ${\ensuremath{\lambda}}_{3}$. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient ${\ensuremath{\lambda}}_{3}$ can be evaluated directly by Euclidean means and does not in general vanish.
Linear response theory relates hydrodynamic transport coefficients to equilibrium retarded correlation functions of the stress-energy tensor and global symmetry currents in terms of Kubo formulas. Some of these transport coefficients are non-dissipative and affect the fluid dynamics at equilibrium. We present an algebraic framework for deriving Kubo formulas for such thermal transport coefficients by using the equilibrium partition function. We use the framework to derive Kubo formulas for all such transport coefficients of superfluids, as well as to rederive Kubo formulas for various normal fluid systems.
Relativistic hydrodynamics is a low-energy e↵ective theory which universally describes macroscopic behaviors of relativistic many-body systems.Its application covers the broad branches of physics such as high-energy physics, astrophysics, and condensed matter physics.Nevertheless, its derivation from underlying microscopic theories, especially from quantum field theories, has not been clearly clarified.Furthermore, it is recently pointed out that novel transport phenomena such as the chiral magnetic e↵ect, which originates from the quantum anomaly in the underlying quantum field theories, take place in the macroscopic hydrodynamic regime.In this thesis, we derive relativistic hydrodynamics from quantum field theories on the basis of the recent development of nonequilibrium statistical mechanics.In order to derive the hydrodynamic equations we introduce an assumption that the density operator is given by a local Gibbs distribution at initial time, and decompose the energy-momentum tensor and charge current into nondissipative and dissipative parts.This leads to a generalization of the Gibbs ensemble method canonically employed in equilibrium statistical mechanics.Our formalism is also applicable to the situation in the presence of the quantum anomaly, and we can describe the anomaly-induced tranport phenomena.As a basic tool of our formalism, we first develop a path integral of the thermodynamic potential for locally thermalized systems.We show microscopically that the thermodynamic pontential, which is shown to be the generating functional of systems in local thermal equilibrium, is written in terms of the quantum field theory in the curved spacetime with one imaginary-time direction.The structure of this thermally emergent curved spacetime is determined by hydrodynamic variables such as the local temperature, and fluid-four velocity, and possesses notable symmetry properties: Kaluza-Klein gauge symmetry, spatial di↵eomorphism symmetry, and gauge symmetry.With the help of the symmetry argument, we can construct the nondissipative part of the hydrodynamic equations including the anomaly-induced transport phenomena.By the use of the perturbative calculation, we evaluate the anomalous transport coe cients at one-loop level.Furthermore, we also construct a solid basis to study dissipative corrections to hydrodynamic equations.In particular, by performing the derivative expansion, together with the result on nondissipative part of the constitutive relations, we derive the firstorder dissipative hydrodynamic equations, that is, the relativistic Navier-Stokes equation.Our formalism also provides the quantum field theoretical expression of the Green-Kubo formulas for transport coe cients.iii The formulation and whole works presented from Chapter 2 to Chapter 4 are based on our original work in collaboration with Yoshimasa Hidaka (RIKEN).The work presented in Chapter 3 is in collaboration with Yoshimasa Hidaka (RIKEN), Tomoya Hayata (RIKEN), and Toshifumi Noumi (Hong Kong University of Science and Technology) [1].1 There is close investigation of priority to the Navier-Stokes equation in Ref. [11].