Causal relativistic hydrodynamics for viscous fluids
Ulrich W Heinz, Huichao Song
备注：
4 pages, 1 figure. Contribution to the Proceedings for Quark Matter 2008, Jaipur, India, Feb. 3-10, 2008, to appear in a special issue of J. Phys. G. Version 2: CERN preprint number corrected and Fig. 1 (right panel) recalculated using full Israel-Stewart equations (see text)
We report on recent results from VISH2+1, a code that solves the relativistic Israel-Stewart equations for causal viscous hydrodynamics for heavy-ion collisions with longitudinal boost invariance. We find that even ``minimal'' shear viscosity eta/s=hbar/(4pi) leads to a large reduction of elliptic flow compared to ideal fluid dynamics. We explore systematically the sensitivity of this reduction to the equation of state, system size, initial conditions, and the microscopic relaxation time in different formulations of the Israel-Stewart equations.
A new formulation of (3+1)-dimensional anisotropic hydrodynamics is presented that accounts nonperturbatively for the large longitudinal-transverse pressure anisotropy and bulk viscous pressure in heavy-ion collisions. The initialization of the anisotropic hydrodynamic stage is discussed, and a comparison to standard viscous hydrodynamics for (0+1)- dimensional Bjorken expansion is presented.
The matter created in relativistic heavy ion collisions is fairly well described by ideal hydrodynamics, and somewhat better described by viscous hydrodynamics. To this point, most viscous calculations have been two-dimensional, based on an assumption of Bjorken boost invariance along the beam axis. Here, first results are presented for a fully three-dimensional viscous model. The model is described and tests of the numerical accuracy of the code are presented. Two- and three-dimensional runs are compared, and modest changes are observed for mid-rapidity observables at the highest RHIC (Relati…
We report recent progress on causal viscous hydrodynamics for relativistic heavy ion collisions. For fixed specific shear viscosity eta/s, uncertainties in the elliptic flow arising from initial conditions, equation of state, bulk viscosity and numerical viscosity, and the treatment of the highly viscous hadronic stage and freeze-out procedure are analysed. A comparison of current viscous hydrodynamic results with experimental data yields a robust upper limit eta/s < 5/(4pi).
The viscosity of the QGP is a presently hotly debated subject. Since its computation from first principles is difficult, it is desirable to try to extract it from experimental data. Viscous hydrodynamics provides a tool that can attack this problem and which may work in regions where ideal hydrodynamics begins to fail. This thesis focuses on viscous hydrodynamics for relativistic heavy ion collisions. We first review the 2nd order viscous equations obtained from different approaches, and then report on the work of the Ohio State University group on setting up the equations for causal viscous h…
The Large Hadron Collider (LHC) experiments have revealed that the predictions of the color glass condensate (CGC) tend to underestimate the multiplicity at mid-rapidity. We develop and estimate a full second-order viscous hydrodynamic model for the longitudinal expansion to find that the CGC rapidity distributions are visibly deformed during the hydrodynamic stage due to the interplay between the entropy production and the entropy flux to forward rapidity. The results indicate the importance of viscous hydrodynamic evolution with non-boost invariant flow for understanding the CGC in terms of …
In an earlier work (arXiv:0808.0953) we established that causal Israel-Stewart viscous hydrodynamics is only accurate in RHIC applications at very low shear viscosities 4 pi eta_s / s < ~ 1.5-2. We show here that the region of applicability is significantly reduced if bulk viscosity plays a role in the dynamics.
In this paper, we propose a method of solving the viscous hydrodynamics order by order in a derivative expansion. In such a method, the zero-order solution is just one of the ideal hydrodynamics. All the other higher order corrections satisfy the same first-order partial differential equations but with different inhomogeneous terms. We take the Bjorken flow as an example to test the validity of our method and present how to deal with the problems about the initial condition and perturbation evolution in our formalism.
One of the most striking results from the Relativistic Heavy Ion Collider is the strong elliptic flow. This review summarizes what is observed and how these results are combined with reasonable theoretical assumptions to estimate the shear viscosity of QCD near the phase transition. A data comparison with viscous hydrodynamics and kinetic theory calculations indicates that the shear viscosity to entropy ratio is surprisingly small, $η/s < 0.4$. The preferred range is $η/s \simeq (1\leftrightarrow 3) \times 1/4π$.
We investigate the causality and the stability of the relativistic viscous magneto-hydrodynamics in the framework of the Israel-Stewart (IS) second-order theory, and also within a modified IS theory which incorporates the effect of magnetic fields in the relaxation equations of the viscous stress. We compute the dispersion relation by perturbing the fluid variables around their equilibrium values. In the ideal magnetohydrodynamics limit, the linear dispersion relation yields the well-known propagating modes: the Alfvén and the magneto-sonic modes.In the presence of bulk viscous pressure, the c…
Bulk viscosity suppresses elliptic flow v_2, as does shear viscosity. It can thus not be neglected when extracting the shear viscosity from elliptic flow data. We here explore uncertainties in the bulk viscous contribution to viscous v_2 suppression that arise from presently uncontrolled uncertainties in the initial value of the bulk viscous pressure and its microscopic relaxation time.
Explicit equations are given for describing the space-time evolution of non-ideal (viscous) relativistic fluids undergoing boost-invariant longitudinal and arbitrary transverse expansion. The equations are derived from the second-order Israel-Stewart approach which ensures causal evolution. Both azimuthally symmetric (1+1)-dimensional and non-symmetric (2+1)-dimensional transverse expansion are discussed. The latter provides the formal basis for the hydrodynamic computation of elliptic flow in relativistic heavy-ion collisions including dissipative effects.
The second order Israel-Stewart-M$\ddot{u}$ller relativistic hydrodynamics has been applied to study the effects of causality on the acoustic oscillation in relativistic fluid. Causal dispersion relations have been derived with non-vanishing shear viscosity, bulk viscosity and thermal conductivity at non-zero temperature and chemical potential. These relations have been used to investigate the fluidity of Quark Gluon Plasma (QGP) at finite temperature ($T$). Results of the first order dissipative hydrodynamics have been obtained as limiting case of the second order theory. The effects of the c…
I review recent and not so recent progress on formulating and numerically implementing a consistent set of relativistic equations which describe the space-time evolution of viscous relativistic fluids without violating causality.
Due to large pressure gradients at early times, standard hydrodynamic model simulations of relativistic heavy-ion collisions do not become reliable until $O(1)$\,fm/$c$ after the collision. To address this one often introduces a pre-hydrodynamic stage that models the early evolution microscopically, typically as a conformal, weakly interacting gas. In such an approach the transition from the pre-hydrodynamic to the hydrodynamic stage is discontinuous, introducing considerable theoretical model ambiguity. Alternatively, fluids with large anisotropic pressure gradients can be handled macroscopic…
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.
We study the role of viscosity in the early stages of relativistic heavy ion collisions. We investigate the role of viscosity on the chemical equilibration of a parton gas. In the presence of viscosity the lifetime of the system is increased. The temperature as well as the parton fugacities evolve more slowly compared to the ideal fluid dynamics.
Assuming that in Au+Au collisions, a baryon free fluid is produced, transverse momentum spectra of identified particles ($π$, $K$, $p$ and $ϕ$), in evolution of ideal and viscous fluid is studied. Hydrodynamic evolution is governed by a lattice based equation of state (EOS), where the confinement-deconfinement transition is a cross-over at $T_{co}$=196 MeV. Ideal or viscous fluid was initialised to reproduce $ϕ$ meson multiplicity in 0-5% Au+Au collisions. Ideal or minimally viscous ($η/s$=0.08) fluid evolution reasonably well explain the transverse momentum spectra of pion's, kaon's, and $ϕ$ …
The (viscous) anisotropic hydrodynamic approach, especially after perturbative inclusion of all residual viscous terms, has been shown to dramatically outperform viscous hydrodynamics in several simplified situations for which exact solutions exist but which share with realistic expansion scenarios the problem of large dissipative currents. We will report on the present status of applying viscous anisotropic hydrodynamics in a highly efficient simulation of the full three-dimensional quark-gluon plasma. Results from accelerated $3{+}1$-dimensional viscous hydrodynamic simulations using graphic…
We would like to formulate relativistic dissipative hydrodynamics for multi-component systems with multiple conserved currents. This is important for analyses of the hot matter created in relativistic heavy ion collisions because particle creations and annihilations of various particle species are frequently taking place there. We show that consistent formulation in such systems involves many non-trivialities, and derive constitutive equations that satisfy Onsager reciprocal relations and describe the systems without ambiguity.
In this dissertation I introduce relativistic heavy ion collisions and describe theoretical approaches to understanding them--in particular, viscous hydrodynamic simulations and investigations of final state interactions. The successful ideal hydrodynamic models of the collisions at the Relativistic Heavy Ion Collider (RHIC) were extended by performing viscous hydrodynamic simulations. This was done by making use of the recently derived full conformally invariant second order relativistic viscous hydrodynamic equations. Results for multiplicity, radial flow and elliptic flow in sqrt{s_NN}=200 …
The stability of the 1+1 dimensional solution of Israel-Stewart theory is investigated. Firstly, the evolution of the temperature and the ratio of the bulk pressure over the equilibrium pressure of the background is explored. Then the stability with linear perturbations is studied by using the Lyapunov direct method. It shows that the shear viscosity may weaken the instability induced by the large peak of bulk viscosity around the phase transition temperature $T_c$.
Identified particle observables from viscous hydrodynamics are sensitive to the fluid-to-particle conversion. Instead of the commonly assumed "democratic" Grad ansatz for phase space corrections $δf$, we utilize corrections calculated from linearized covariant transport theory. Estimates based on a pion-proton system with binary collisions indicate that protons are much closer to equilibrium than pions, significantly affecting the dissipative reduction of differential elliptic flow in Au+Au at RHIC. In addition, we test linear response against fully nonlinear transport for a two-component mass…
It has been over a decade since the first experimental data from gold nuclei collisions at the Relativistic Heavy Ion Collider suggested hydrodynamic behavior. While early ideal hydrodynamical models were surprisingly accurate in their predictions, they ignored that the large longitudinal velocity gradient meant that even small shear viscosity would produce large corrections to the transverse dynamics. In addition, much less was known about the equation of state predicted by lattice calculations of quantum chromodynamics, which predicts a soft region as the degrees of freedom change from quark…
Models of relativistic heavy ion collisions typically involve both a hydrodynamic module to describe the high density liquid-like phase and a Boltzmann module to simulate the low density break-up phase which is gas-like. Coupling the prescriptions is more complicated for viscous prescriptions if one wants to maintain continuity of the entire stress-energy tensor and currents. Derivations for the viscosity for a gas are reviewed, which then lead to expressions for changes in the phase space occupation based on simple relaxation-time pictures of viscosity. These expressions are shown to consiste…
The particle spectra and Hanbury-Brown Twiss (HBT) radius of Au+Au collisions at RHIC energy are investigated by a hydrodynamical expanding source with both shear and bulk viscosities ($ζ$). With a large width of the ratio of $ζ$ to entropy density $s$, both the particle transverse momentum spectra and the ratio $R_{out}/R_{side}$ of HBT radii in the direction of the total transverse momentum of detected two particles ($R_{out}$) and perpendicular to both this direction and the beam direction ($R_{side}$) become a little steeper.
Using numerical results from ideal and viscous relativistic hydrodynamic simulations with three different equations of state, for Au+Au and Cu+Cu collisions at different centralities and initial energy densities, we explore the dependence of the eccentricity-scaled elliptic flow, v_2/epsilon, and the produced entropy fraction, Delta S/S_0, on the final charged hadron multiplicity density dN_ch/dy per unit transverse overlap area S, (1/S)(dN_ch/dy). The viscous hydrodynamic simulations are performed with two different versions of the Israel-Stewart kinetic evolution equations, and in each case …
We report part of our recent work on viscous hydrodynamics with consistent phase space distribution $f(x,\p)$ for freeze out. We develop the gradient expansion formalism based on kinetic theory, and with the constraints from the comparison between hydrodynamics and kinetic theory, viscous corrections to $f(x,\p)$ can be consistently determined order by order. Then with the obtained $f(x,\p)$, second order viscous hydrodynamical calculations are carried out for elliptic flow $v_2$.
We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic an…
We show that by requiring positivity of the longitudinal pressure it is possible to constrain the initial conditions one can use in 2nd-order viscous hydrodynamical simulations of ultrarelativistic heavy-ion collisions. We demonstrate this explicitly for 0+1 dimensional viscous hydrodynamics and discuss how the constraint extends to higher dimensions. Additionally, we present an analytic approximation to the solution of 0+1 dimensional 2nd-order viscous hydrodynamical evolution equations appropriate to describe the evolution of matter in an ultrarelativistic heavy-ion collision.
Viscous corrections to relativistic hydrodynamics, which are usually formulated for small velocity g radients, have recently been extended from Navier-Stokes formulations to a class of treatments based on Israel-Stewart equations. Israel-Stewart treatments, which treat the spatial components of the s tress-energy tensor tau_ij as dynamical objects, introduce new parameters, such as the relaxati on times describing non-equilibrium behavior of the elements tau_ij. By considering linear resp onse theory and entropy constraints, we show how the additional parameters are related to fluctuatio ns of…
We compare 2->2 covariant transport theory and causal Israel-Stewart hydrodynamics in 2+1D longitudinally boost invariant geometry with RHIC-like initial conditions and a conformal e = 3p equation of state. The pressure evolution in the center of the collision zone and the final differential elliptic flow v2(pT) from the two theories agree remarkably well for a small shear viscosity to entropy density ratio eta/s ~ 1/(4 pi), and also for a large cross section sigma ~ 50 mb. A key to this agreement is keeping ALL terms in the Israel-Stewart equations of motion. Our results indicate promising pr…
We extend our hybrid model HydHSD by taking into account shear viscosity within the Israel-Stewart hydrodynamics. The influence of different forms of $π^{μν}$ constraints on observables is analyzed. We show that the form of the corresponding condition plays an important role for the sensitivity of viscous hydrodynamics to the ratio of shear viscosity to the entropy density, $η/s$. It is shown that the constraint used in the vHLLE model, results in most sensitivity of rapidity distributions and transverse momentum spectra to a change of the $η/s$ ratio; however, their applicability for large va…
We present a first calculation of the dilepton yield and elliptic flow done with 3+1D viscous hydrodynamical simulations of relativistic heavy ion collisions at the top RHIC energy. A comparison with recent experimental data from the STAR collaboration is made.
We investigate effects of causal hydrodynamic fluctuations in the longitudinally expanding quark gluon plasma on final entropy distributions in high-energy nuclear collisions.
We study the theory of relativistic viscous hydrodynamics introduced in arXiv:1109.0985 and arXiv:1907.12695, which provided a causal and stable first-order theory of relativistic fluids with viscosity in the case of barotropic fluids. The local well-posedness of its equations of motion has been previously established in Gevrey spaces. Here, we improve this result by proving local well-posedness in Sobolev spaces.
Extended applicability domain of viscous anisotropic hydrodynamics in (2+1)-D Bjorken flow with transverse expansion
Peking University（北京大学）；Ghent University（根特大学）；Universität Bielefeld（比勒费尔德大学）；The Ohio State University（俄亥俄州立大学）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
Using the (2+1)-dimensional ultrarelativistic viscous hydrodynamics+hadron cascade, VISHNU, hybrid model, we study the $p_{\rm T}$-spectra and elliptic flow of $Λ$, $Ξ$, and $Ω$ in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV and in Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV. Comparing our model results with the data measurements, we find that the VISHNU model gives general descriptions of the measurements of these strange and multi-strange hadrons at several centrality classes at RHIC and LHC. Mass ordering of elliptic flow $v_{2}$ among $π$, $K$, $p$, $Λ$, $Ξ$, and $Ω$ are further investiga…
Hydrodynamic model simulations of Au-Au collisions at RHIC have indicated recently, that with improved simulations in the coming years, it may be feasible to quantify the viscosity of the matter produced in heavy ion collisions. To this end, a consistent fluid dynamic description of viscous effects is clearly needed. In this note, we observe that a recently used, approximate form of the 2nd order Israel-Stewart viscous hydrodynamic equations of motion cannot account consistently for the transverse flow fields, which are expected to develop in heavy ion collisions. We identify the appropriate e…
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-lo…
We generalize the derivation of viscous anisotropic hydrodynamics from kinetic theory to allow for non-zero particle masses. The macroscopic theory is obtained by taking moments of the Boltzmann equation after expanding the distribution function around a spheroidally deformed local momentum distribution whose form has been generalized by the addition of a scalar field that accounts non-perturbatively (i.e. already at leading order) for bulk viscous effects. Hydrodynamic equations for the parameters of the leading-order distribution function and for the residual (next-to-leading order) dissipat…
Using ideal and viscous hydrodynamics, the ratio of azimuthal moments v4/(v2)^2 is calculated for pions, protons, and kaons in sqrt{s}=200 A*GeV Au+Au collisions. For any value of viscosity here is little dependence on particle species. Ideal hydrodynamics and data show a flat curve as a function of pt. Adding viscosity in the standard way destroys this flatness. However, it can be restored by replacing the standard quadratic ansatz for delta f (the viscous correction to the distribution function at freeze-out) with a weaker momentum dependence.
We study causal viscous hydrodynamics in the context of central relativistic heavy-ion collisions and provide details of a straightforward numerical algorithm to solve the hydrodynamic equations. It is shown that correlation functions of fluctuations provide stringent test cases for any such numerical algorithm. Passing these tests, we study the effects of viscosity on the temperature profile in central heavy-ion collisions. Also, we find that it is possible to counter-act the effects of viscosity to some extent by re-adjusting the initial conditions. However, viscous corrections are strongest…
We describe the details of 3+1 dimensional relativistic hydrodynamic code for the simulations of quark-gluon/hadron matter expansion in ultra-relativistic heavy ion collisions. The code solves the equations of relativistic viscous hydrodynamics in the Israel-Stewart framework. With the help of ideal-viscous splitting, we keep the ability to solve the equations of ideal hydrodynamics in the limit of zero viscosities using a Godunov-type algorithm. Milne coordinates are used to treat the predominant expansion in longitudinal (beam) direction effectively. The results are successfully tested again…
We investigate hydrodynamic evolution of the quark gluon plasma for the colour glass condensate type initial conditions. We solve full second-order viscous hydrodynamic equations in the longitudinal direction to find that non-boost invariant expansion leads to visible deformation on the initial rapidity distribution. The results indicate that hydrodynamic evolution with viscosity plays an important role in determining parameters for the initial distributions.
The expansion of the fireball created in Au-Au collisions at 200 GeV is described in (3+1)-dimensional viscous hydrodynamics with shear and bulk viscosities. We present results for the transverse momentum spectra, the directed and elliptic flow and the interferometry radii.
A novel formulation of second-order relativistic viscous fluid dynamics based on the effective Boltzmann equation for quasi-particles with medium-dependent masses is briefly reviewed.~The evolution equations for the shear and bulk dissipative corrections, and the corresponding transport coefficients, are presented. Resulting approach allows for thermodynamically-consistent incorporation of lattice QCD equation of state in the fluid dynamical framework.
Hydrodynamics is a general theoretical framework for describing the long-time large-distance behaviors of various macroscopic physical systems, with its equations based on conservation laws such as energy-momentum conservation and charge conservation. Recently there has been significant interest in understanding the implications of angular momentum conservation for a corresponding hydrodynamic theory. In this work, we examine the key conceptual issues for such a theory in the relativistic regime where the orbital and spin components get entangled. We derive the equations for relativistic visco…
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 ($v_2$, $v_3$ and $v_4$) and momentum spectrum using a transport calculation. The linear response coefficient to the initial an…
The invariant mass spectrum and the elliptic flow of lepton pairs produced in relativistic heavy-ion collisions at RHIC are studied with viscous hydrodynamics. The effects of viscous corrections on dilepton observables are explored. The lepton pairs originating from charm quarks evolving in the viscous background are seen to be a good probe of quark energy loss and gain, as quantified by the dilepton spectrum and by the dilepton elliptic flow.
We present the first calculation of longitudinal decorrelation of anisotropic flow at RHIC Beam Energy Scan (BES) energies using event-by-event viscous hydrodynamic model (vHLLE), with two different initial states (GLISSANDO2 and UrQMD) and hadronic cascade. We investigate the origin of the observed decorrelation by checking separately flow angle and flow magnitude decorrelation and by calculating decorrelation in the initial state eccentricity.
Causal viscous hydrodynamic fits to experimental data for pion and kaon transverse momentum spectra from central Au+Au collisions at \sqrt{s_{NN}}=200 GeV are presented. Starting the hydrodynamic evolution at 1 fm/c and using small values for the relaxation time, reasonable fits up to moderate ratios η/s\simeq 0.4 can be obtained. It is found that a percentage of roughly 50 η/s to 75 η/s of the final meson multiplicity is due to viscous entropy production. Finally, it is shown that with increasing viscosity, the ratio of HBT radii R_{out}/R_{side} approaches and eventually matches the experime…
Event-by-event hydrodynamics is applied to Cu-Au collisions at 200GeV. Predictions for charged particle distributions in pseudorapidity, transverse momentum spectra, femtoscopy radii are given. The triangular and elliptic flow coefficients are calculated. The directed flow at central rapidity in the reaction plane in asymmetric collisions is nonzero, fluctuations of the initial profile lead to a further increase of the directed flow when measured in the event plane.
This is a comment to hep-th/0703243. The paper determined the relaxation time tau_pi of the shear viscous stress for the N=4 SYM from AdS/CFT correspondence. The purpose of this comment is to point out that the value of tau_pi is 3 times larger than their result if one takes into account an additional term in the hydrodynamic equation.
We investigate the phenomenology of freely expanding fluids, with different material properties, evolving through the Israel-Stewart (IS) causal viscous hydrodynamics, and compare our results with those obtained in the relativistic Eckart-Landau-Navier-Stokes (ELNS) acausal viscous hydrodynamics. Through the analysis of scaling invariants we give a definition of thermalization time which can be self-consistently determined in viscous hydrodynamics. Next we construct the solutions for one-dimensional boost-invariant flows. Expansion of viscous fluids is slower than that of one-dimensional ideal…
A new formulation of second-order viscous hydrodynamics, based on an expansion around a locally anisotropic momentum distribution, is presented. It generalizes the previously developed formalism of anisotropic hydrodynamics (aHydro) to include a complete set of dissipative currents for which equations of motion are derived by solving the Boltzmann equation in the 14-moment approximation. By solving the vaHydro equations for a transversally homogeneous, longitudinally boost-invariant system ((0+1)-dimensional expansion) and comparing with the exact solution of the Boltzmann equation in relaxati…
