Second-order Viscous Hydrodynamics Research Articles

Hydrodynamization and resummed viscous hydrodynamics

In this paper, I review our current understanding of the applicability of hydrodynamics to modeling the quark–gluon plasma (QGP), focusing on the question of hydrodynamization/thermalization of the QGP and the anisotropic hydrodynamics (aHydro) far-from-equilibrium hydrodynamic framework. I discuss the existence of far-from-equilibrium hydrodynamic attractors and methods for determining attractors within different hydrodynamical frameworks. I also discuss the determination of attractors from exact solutions to the Boltzmann equation in relaxation time approximation and effective kinetic field theory applied to quantum chromodynamics. I then present comparisons of the kinetic attractors with the attractors obtained in standard second-viscous hydrodynamics frameworks and aHydro. I demonstrate that, due to the resummation of terms to all orders in the inverse Reynolds number, the aHydro framework can describe both the weak- and strong-interaction limits. I then review the phenomenological application of aHydro to relativistic heavy-ion collisions using both quasiparticle aHydro and second-order viscous aHydro. The phenomenological results indicate that aHydro provides a controlled extension of dissipative relativistic hydrodynamics to the early-time far-from-equilibrium stage of heavy-ion collisions. This allows one to better describe the data and to extract the temperature dependence of transport coefficients at much higher temperatures than linearized second-order viscous hydrodynamics.

Relativistic second-order viscous hydrodynamics from kinetic theory with extended relaxation-time approximation

Relativistic second-order viscous hydrodynamics from kinetic theory with extended relaxation-time approximation

Impact of vorticity and viscosity on the hydrodynamic evolution of hot QCD medium

The strongly interacting transient state of quark-gluon plasma (QGP) medium created in ultra-relativistic collisions survives for a duration of a few fm/c. The spacetime evolution of QGP crucially depends on the equation of state (EoS), vorticity, viscosity, and external magnetic field. In the present study, we obtain the lifetime of a vortical QGP fluid within the ambit of relativistic second-order viscous hydrodynamics. We observe that the coupling of vorticity and viscosity significantly increases the lifetime of vortical QGP. The inclusion of a static magnetic field, vorticity, and viscosity makes the evolution slower. However, the static magnetic field slightly decreases the QGP lifetime by accelerating the evolution process for a non-rotating medium. We also report the rate of change of vorticity in the QGP, which will be helpful in studying the behavior of the medium in detail.

Impact of fluctuating initial conditions on bottomonium suppression in 5.02 TeV heavy-ion collisions

We compute bottomonium suppression and elliptic flow within the pNRQCD effective field theory using an open quantum systems approach. For the hydrodynamical background, we use 2+1D MUSIC second-order viscous hydrodynamics with IP-Glasma initial conditions and evolve bottom/antibottom quantum wave packets in real time in these backgrounds. We find that the impact of fluctuating initial conditions is small when compared to results obtained using smooth initial conditions. Including the effect of fluctuating initial conditions, we find that the Upsilon(1S) integrated elliptic flow is v_2[1S] = 0.005 +/- 0.002 +/- 0.001, with the first and second variations corresponding to statistical and systematic theoretical uncertainties, respectively.

Transient Relativistic Fluid Dynamics in a General Hydrodynamic Frame.

We propose a new theory of second-order viscous relativistic hydrodynamics which does not impose any frame conditions on the choice of the hydrodynamic variables. It differs from Mueller-Israel-Stewart theory by including additional transient degrees of freedom, and its first-order truncation reduces to Bemfica-Disconzi-Noronha-Kovtun theory. Conditions for causality and stability are explicitly given in the conformal regime. As an illustrative example, we consider Bjorken flow solutions to our equations and identify variables which make a hydrodynamic attractor manifest.

Nonconformal kinetic theory and hydrodynamics for Bjorken flow

Using and comparing kinetic theory and second-order Chapman-Enskog hydrodynamics, we study the non-conformal dynamics of a system undergoing Bjorken expansion. We use the concept of `free-streaming fixed lines' for scaled shear and bulk stresses in non-conformal kinetic theory and hydrodynamics, and show that these `fixed lines' behave as early-time attractors and repellors of the evolution. In the conformal limit, the free-streaming fixed lines reduce to the well-known fixed points of conformal Bjorken dynamics. A new fixed point in the free streaming regime is identified which lies at the intersection of these fixed lines. Contrary to the conformal scenario, both kinetic theory and hydrodynamics predict the absence of attractor behavior in the normalised shear stress channel. In kinetic theory a far-off-equilibrium attractor is found for the normalised effective longitudinal pressure, driven by rapid longitudinal expansion. Second-order viscous hydrodynamics fails to accurately describe this attractor. From a thorough analysis of the free-streaming dynamics in Chapman-Enskog hydrodynamics we conclude that this failure results from an inaccurate approximation of the fixed lines and a related incorrect description of the nature of the fixed point. A modified anisotropic hydrodynamic description is presented that provides excellent agreement with kinetic theory results and reproduces the far-from-equilibrium attractor for the scaled longitudinal pressure.

Hydrodynamic attractor in the nonconformal Bjorken flow

Nonconformal attractor behavior is studied by solving nonconformal second-order viscous hydrodynamics with respect to boost-invariant plasmas. Numerical solutions of the relative decay rate of the enthalpy density, the inverse shear and bulk Reynolds numbers all exhibit universal patterns towards an ideal fluid description at late stages, demonstrating the existence of a nonconformal hydrodynamic attractor. However, as a generic feature, the nonconformal hydrodynamic attractor emerges only at late times, which differs drastically from the behavior observed in conformal systems. The absence of the early-time attractor in nonconformal fluids is a consequence of the early-time unstable mode, which arises from a positive-valued eigenmode associated with the free-streaming dynamics. The early-time attractor can be restored in a mixture of the shear and bulk viscous corrections, in which early-time unstable modes cancel approximately. However, to support practical simulations of hydrodynamics in high-energy heavy-ion collisions, for which the early-time nonconformal attractor is approachable in all independent modes, the quadratic coupling between the bulk pressure and the expansion rate should be enhanced by at least a factor of two through the transport coefficient ${\ensuremath{\delta}}_{\mathrm{\ensuremath{\Pi}}\mathrm{\ensuremath{\Pi}}}$.

Attractor and fixed points in Bjorken flows

We consider a plasma of massless particles undergoing Bjorken expansion, mimicking the matter created in ultrarelativistic heavy ion collisions. We study the transition to hydrodynamics using kinetic theory in the relaxation time approximation. By allowing the relaxation time to depend on time, we can monitor the speed of the transition from the collisionless regime to hydrodynamics. By using a special set of moments of the momentum distribution, we reduce the kinetic equation to a coupled mode problem which encompasses all versions of second-order viscous hydrodynamics for Bjorken flows. This coupled mode problem is analyzed first using techniques of linear algebra. Then we transform this two-mode problem into a single nonlinear differential equation and proceed to a fixed point analysis. We identify an attractor solution as the particular solution of this nonlinear equation that joins two fixed points: one corresponding to the collisionless, early time regime, the other corresponding to late time hydrodynamics. We exploit the analytic solution of this equation in order to test several approximations and to identify generic features of the transition to hydrodynamics. We argue that extending the accuracy of hydrodynamics to early time, i.e., to the region of large gradients, amounts essentially to improve the accuracy of the location of the collisionless fixed point. This is demonstrated by showing that a simple renormalization of a second-order transport coefficient puts the free streaming fixed point at the right location, and allows us to reproduce accurately the full solution of the kinetic equation within second-order viscous hydrodynamics, even in regimes far from local equilibrium.

Second order relativistic viscous hydrodynamics within an effective description of hot QCD medium

The second-order hydrodynamic equations for evolution of shear and bulk viscous pressure have been derived within the framework of covariant kinetic theory based on the effective fugacity quasiparticle model. The temperature-dependent fugacity parameter in the equilibrium distribution function leads to a mean field term in the Boltzmann equation which affects the interactions in the hot QCD matter. The viscous corrections to distribution function, up to second-order in gradient expansion, have been obtained by employing a Chapman–Enskog like iterative solution of the effective Boltzmann equation within the relaxation time approximation. The effect of mean field contributions to transport coefficients as well as entropy current has been studied up to second-order in gradients. In contrast to the previous calculations, we find non-vanishing entropy flux at second order. The effective description of relativistic second-order viscous hydrodynamics, for a system of interacting quarks and gluons, has been quantitatively analyzed in the case of the 1 + 1-dimensional boost invariant longitudinal expansion. We study the proper time evolution of temperature, pressure anisotropy, and viscous corrections to entropy density for this simplified expansion. The second order evolution of quark–gluon plasma is seen to be affected significantly with the inclusion of mean field contributions and the realistic equation of state.

Anisotropic fluid dynamical simulations of heavy-ion collisions

We present VAH, a (3+1)–dimensional simulation that evolves the far-from-equilibrium quark-gluon plasma produced in ultrarelativistic heavy-ion collisions with anisotropic fluid dynamics. We solve the hydrodynamic equations on an Eulerian grid using the Kurganov–Tadmor algorithm in combination with a new adaptive Runge–Kutta method. Our numerical scheme allows us to start the simulation soon after the nuclear collision, largely avoiding the need to integrate it with a separate pre-equilibrium dynamics module. We test the code's performance by simulating on the Eulerian grid conformal and non-conformal Bjorken flow as well as conformal Gubser flow, whose (0+1)–dimensional solutions are precisely known. Finally, we compare non-conformal anisotropic hydrodynamics to second-order viscous hydrodynamics in central Pb+Pb collisions and find that the former's longitudinal flow profile responds more consistently to the fluid's gradients along the spacetime rapidity direction. Program summaryManuscript Title: Anisotropic fluid dynamical simulations of heavy-ion collisionsAuthors: Mike McNelis, Dennis Bazow, Ulrich HeinzProgram Title:VAHLicensing provisions: GPLv3Programming Language: C++Computer: Laptop, desktop, clusterOperating System: GNU/Linux distributions, Mac OS XGlobal memory usage: 1.2 GB (for a 129×129×63 grid)Keywords: Ultrarelativistic heavy-ion collisions, quark-gluon plasma, relativistic hydrodynamics, computational fluid dynamicsClassification: 12 Gases and Fluids, 17 Nuclear physicsExternal routines/libraries: GNU Scientific Library (GSL)Nature of problem: Modeling the far-from-equilibrium dynamics of quark-gluon plasma produced in ultrarelativistic heavy-ion collisions.Solution method: Kurganov–Tadmor algorithm, adaptive stepsize methodRunning time: A (3+1)–d non-conformal anisotropic fluid dynamical simulation of a central Pb+Pb collision on a 129×129×63 grid takes about 530s for an Intel Xeon E5-2680 v4 multi-core processor with OpenMP acceleration (see Sec. 5.3).

Thermal noise in non-boost-invariant dissipative hydrodynamics

We study the effects of hydrodynamic fluctuations in non-boost-invariant longitudinal expansion of matter formed in relativistic heavy ion collisions. We formulate the theory of thermal noise within second-order viscous hydrodynamics treating noise as a perturbation on top of the non-boost-invariant flow. We develop a numerical simulation model to treat the (1+1)-dimension hydrodynamic evolution. The code is tested to reproduce the analytic results for the Riemann solver for expansion of matter in vacuum. For viscous hydrodynamic expansion, the initial energy density distribution are obtained by reproducing the measured charged hadron rapidity distribution at the RHIC energies. We show that the longitudinal rapidity correlations arising from space-time dependent thermal noise and from an induced thermal perturbation have distinct structures. In general, the rapidity correlations are found to be dominated by temperature fluctuations at small rapidity separation and velocity fluctuations at large rapidities. We demonstrate that thermal noise produce ridge-like two-particle rapidity correlations which persist at moderately large rapidities. The magnitude and pattern of the correlations are quite sensitive to various second-order dissipative formalisms and to the underlying equations of state, especially at large rapidities. The short-range part of the rapidity correlation is found to be somewhat enhanced as compared to that in boost-invariant flow of matter.

Quasiparticle second-order viscous hydrodynamics from kinetic theory

We present the derivation of second-order relativistic viscous hydrodynamics from an effective Boltzmann equation for a system consisting of quasiparticles of a single species. We consider temperature-dependent masses of the quasiparticles and devise a thermodynamically-consistent framework to formulate second-order evolution equations for shear and bulk viscous pressure corrections. The main advantage of this formulation is that one can consistently implement realistic equation of state of the medium within the framework of kinetic theory. Specializing to the case of one-dimensional purely-longitudinal boost-invariant expansion, we study the effect of this new formulation on viscous hydrodynamic evolution of strongly-interacting matter formed in relativistic heavy-ion collisions.

Non-boost-invariant dissipative hydrodynamics

The one-dimensional non-boost-invariant evolution of the quark-gluon plasma, presumably produced during the early stages of heavy-ion collisions, is analyzed within the frameworks of viscous and anisotropic hydrodynamics. We neglect transverse dynamics and assume homogeneous conditions in the transverse plane but, differently from Bjorken expansion, we relax longitudinal boost invariance in order to study the rapidity dependence of various hydrodynamical observables. We compare the results obtained using several formulations of second-order viscous hydrodynamics with a recent approach to anisotropic hydrodynamics, which treats the large initial pressure anisotropy in a non-perturbative fashion. The results obtained with second-order viscous hydrodynamics depend on the particular choice of the second-order terms included, which suggests that the latter should be included in the most complete way. The results of anisotropic hydrodynamics and viscous hydrodynamics agree for the central hot part of the system, however, they differ at the edges where the approach of anisotropic hydrodynamics helps to control the undesirable growth of viscous corrections observed in standard frameworks.

Highly-anisotropic Hydrodynamics for Central Collisions

The framework of leading-order anisotropic hydrodynamics is supplemented with realistic equation of state and self-consistent freeze-out prescription. The model is applied to central proton-nucleus collisions. The results are compared to those obtained within standard Israel-Stewart second-order viscous hydrodynamics. It is shown that the resulting hadron spectra are highly-sensitive to the hydrodynamic approach that has been used.

Testing different formulations of leading-order anisotropic hydrodynamics

A recently obtained set of the equations for leading-order (3+1)D anisotropic hydrodynamics is tested against exact solutions of the Boltzmann equation with the collisional kernel treated in the relaxation time approximation. In order to perform detailed comparisons, the new anisotropic hydrodynamics equations are reduced to the boost-invariant and transversally homogeneous case. The agreement with the exact solutions found using the new anisotropic hydrodynamics equations is similar to that found using previous, less general formulations of anisotropic hydrodynamics. In addition, we find that, when compared to a state-of-the-art second-order viscous hydrodynamics framework, leading-order anisotropic hydrodynamics better reproduces the exact solution for the pressure anisotropy and gives comparable results for the bulk pressure evolution. Finally, we compare the transport coefficients obtained using linearized anisotropic hydrodynamics with results obtained using second-order viscous hydrodynamics.

Leading-order anisotropic hydrodynamics for central collisions

We use leading-order anisotropic hydrodynamics to study an azimuthally-symmetric boost-invariant quark-gluon plasma. We impose a realistic lattice-based equation of state and perform self-consistent anisotropic freeze-out to hadronic degrees of freedom. We then compare our results for the full spatiotemporal evolution of the quark-gluon plasma and its subsequent freeze-out to results obtained using 1+1d Israel-Stewart second-order viscous hydrodynamics. We find that for small shear viscosities, 4 pi eta/s ~ 1, the two methods agree well for nucleus-nucleus collisions, however, for large shear viscosity to entropy density ratios or proton-nucleus collisions we find important corrections to the Israel-Stewart results for the final particle spectra and the total number of charged particles. Finally, we demonstrate that the total number of charged particles produced is a monotonically increasing function of 4 pi eta/s in Israel-Stewart viscous hydrodynamics whereas in anisotropic hydrodynamics it has a maximum at 4 pi eta/s ~ 10. For all 4 pi eta/s > 0, we find that for Pb-Pb collisions Israel-Stewart viscous hydrodynamics predicts more dissipative particle production than anisotropic hydrodynamics.

(3 + 1)-dimensional framework for leading-order nonconformal anisotropic hydrodynamics

In this work I develop a new framework for anisotropic hydrodynamics that generalizes the leading order of the hydrodynamic expansion to the full (3+1)-dimensional anisotropic massive case. Following previous works, my considerations are based on the Boltzmann kinetic equation with the collisional term treated in the relaxation time approximation. The momentum anisotropy is included explicitly in the leading term, allowing for a large difference between the longitudinal and transverse pressures as well as for non trivial transverse dynamics. Energy and momentum conservation is expressed by the first moment of the Boltzmann equation. The system of equations is closed by using the zeroth and second moments of the Boltzmann equation. The close-to-equilibrium matching with second-order viscous hydrodynamics is demonstrated. In particular, I show that the coupling between shear and bulk pressure corrections, recently proved to be important for an accurate description of momentum anisotropy and bulk viscous dynamics, does not vanish in the close-to-equilibrium limit.

Viscous corrections to anisotropic flow and transverse momentum spectra from transport theory

Viscous hydrodynamics is commonly used to model the evolution of the matter created in an ultra-relativistic heavy-ion collision. It provides a good description of transverse momentum spectra and anisotropic flow. These observables, however, cannot be consistently derived using viscous hydrodynamics alone, because they depend on the microscopic interactions at freeze-out. We derive the ideal hydrodynamic limit and the first-order viscous correction to anisotropic flow (v2, v3 and v4) and momentum spectrum using a transport calculation. The linear response coefficient to the initial anisotropy, vn(pT)/εn, depends little on n in the ideal hydrodynamic limit. The viscous correction to the spectrum depends not only on the differential cross section, but also on the initial momentum distribution. This dependence is not captured by standard second-order viscous hydrodynamics. The viscous correction to anisotropic flow increases with pT, but this increase is slower than usually assumed in viscous hydrodynamic calculations. In particular, it is too slow to explain the observed maximum of vn at pT∼3 GeV/c.

Relativistic quantum transport coefficients for second-order viscous hydrodynamics

We express the transport coefficients appearing in the second-order evolution equations for bulk viscous pressure and shear stress tensor using Bose-Einstein, Boltzmann, and Fermi-Dirac statistics for the equilibrium distribution function and Grad's 14-moment approximation as well as the method of Chapman-Enskog expansion for the non-equilibrium part. Specializing to the case of transversally homogeneous and boost-invariant longitudinal expansion of the viscous medium, we compare the results obtained using the above methods with those obtained from the exact solution of the massive 0+1d relativistic Boltzmann equation in the relaxation-time approximation. We show that compared to the 14-moment approximation, the hydrodynamic transport coefficients obtained by employing the Chapman-Enskog method leads to better agreement with the exact solution of the relativistic Boltzmann equation.

Thermally fluctuating second-order viscous hydrodynamics and heavy-ion collisions

The fluctuation-dissipation theorem requires the presence of thermal noise in viscous fluids. The time and length scales of heavy ion collisions are small enough so that the thermal noise can have a measurable effect on observables. Thermal noise is included in numerical simulations of high energy lead-lead collisions, increasing average values of the momentum eccentricity and contributing to its event by event fluctuations.