In this paper, we perform a linear stability analysis of Israel-Stewart theory around a global equilibrium state, including the effects of shear-stress tensor, net-baryon diffusion current and diffusion-viscous coupling. We find all the relevant modes of this theory and derive necessary conditions that these modes must satisfy in order to be stable and subluminal. With these conditions, we then derive constraints for the shear and diffusion relaxation times and the transport coefficients related to diffusion-viscous coupling.
We rederive relativistic hydrodynamics as a Lagrangian effective theory using the doubled coordinates technique, allowing us to include dissipative terms. We include Navier-Stokes shear and bulk terms, as well as Israel-Stewart relaxation time terms, within this formalism. We show how the inclusion of shear viscosity, and the requirement of a bounded energy-momentum "vacuum", forces the inclusion of the Israel-Stewart term into the theory, thereby providing a justification for the origin and uniqueness of these terms.
The Quark Gluon Plasma produced in heavy-ion collisions has three relevant conserved charges: baryon number (B), strangeness (S), and electric charge (Q). Here we derive the Israel-Stewart framework for BSQ diffusion coupled to shear and bulk viscosity and the thermodynamic derivatives needed to couple this to a BSQ equation of state. We reproduce a subset of the ${\mathcal{J}}$ terms in the DNMR approach and propose a technique to derive other terms from the maximum entropy principle. Finally, we preform a stability analysis that can be used to constraint transport coefficients in BSQ hydrody…
We show that the recently formulated causal and stable first-order hydrodynamics has the same dynamics as Israel-Stewart theory for boost-invariant, Bjorken expanding systems with a conformal equation of state and a regulating sector determined by a constant relaxation time. In this case, the general solution of the new first-order formulation can be determined analytically.
Exact correspondence between Israel-Stewart theory and first-order causal and stable hydrodynamics is established for the boost-invariant massive case with zero baryon density and the same constant relaxation times used in the shear and bulk sectors. Explicit expressions for the temperature dependent regulators are given for the case of a relativistic massive gas. The stability and causality conditions known in the first-order approach are applied and one finds that one of them is violated in this case.
We obtain an exact correspondence between the dynamical equations in Israel-Stewart (IS) theory and first-order causal and stable (FOCS) hydrodynamics for a boost-invariant system with an ideal gas equation of state at finite baryon chemical potential. Explicit expressions for the temperature and chemical potential dependence of the regulators in the FOCS theory are given in terms of the kinetic coefficients and constant relaxation time of the IS theory. Using the correspondence between the IS and FOCS theory, stability conditions for a charged fluid which are known in the FOCS approach are ap…
We use symmetry arguments developed by Gubser to construct the first radially-expanding explicit solutions of the Israel-Stewart formulation of hydrodynamics. Along with a general semi-analytical solution, an exact analytical solution is given which is valid in the cold plasma limit where viscous effects from shear viscosity and the relaxation time coefficient are important. The radially expanding solutions presented in this paper can be used as nontrivial checks of numerical algorithms employed in hydrodynamic simulations of the quark-gluon plasma formed in ultra-relativistic heavy ion collis…
The solutions of relativistic viscous hydrodynamics for longitudinal expanding fireballs is investigated with the Navier-Stokes theory and Israel-Stewart theory. The energy and Euler conservation equations for the viscous fluid are derived in Rindler coordinates with the longitudinal expansion effect is small. Under the perturbation assumption, an analytical perturbation solution for the Navier-Stokes approximation and numerical solutions for the Israel-Stewart approximation are presented. The temperature evolution with both shear viscous effect and longitudinal acceleration effect in the long…
We prove that Einstein's equations coupled to equations of Israel-Stewart-type, describing the dynamics of a relativistic fluid with bulk viscosity and nonzero baryon charge (without shear viscosity or baryon diffusion) dynamically coupled to gravity, are causal in the full nonlinear regime. We also show that these equations can be written as a first-order symmetric hyperbolic system, implying local existence and uniqueness of solutions to the equations of motion. We use an arbitrary equation of state and do not make any simplifying symmetry or near-equilibrium assumption, requiring only physi…
We utilize nonequilibrium covariant transport theory to determine the region of validity of causal Israel-Stewart dissipative hydrodynamics (IS) and Navier-Stokes theory (NS) for relativistic heavy ion physics applications. A massless ideal gas with 2->2 interactions is considered in a 0+1D Bjorken scenario, appropriate for the early longitudinal expansion stage of the collision. In the scale invariant case of a constant shear viscosity to entropy density ratio eta/s ~ const, we find that Israel-Stewart theory is 10% accurate in calculating dissipative effects if initially the expansion timesc…
We construct a relativistic model for bulk viscosity and heat conduction in a superfluid. Building on the principles of Unified Extended Irreversible Thermodynamics, the model is derived from Carter's multifluid approach for a theory with 3 four-currents: particles, entropy, and quasi-particles. Dissipation arises directly from the fact that the quasi-particle four-current is an independent degree of freedom that does not necessarily comove with the entropy. For small deviations from local thermodynamic equilibrium, the model provides an extension of the Israel-Stewart theory to superfluid sys…
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…
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…
Nonlinear causality of Israel-Stewart theory with diffusion
Department of Mathematics, Vanderbilt University（范德堡大学数学系）；Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学科学与技术学院）；Department of Physics and Astronomy, University of Florence（佛罗伦萨大学物理与天文学系）；Theoretical Physics Department, CERN（欧洲核子研究中心理论物理部）
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 …
We investigate the implications of the general frame approach for conformal Bjorken flow beyond the earlier studies. We show that the power series solution at late times is not unique and is accompanied by an exact solution of the form $1/τ$, which becomes unphysical if taken on shell. In contrast to the Müller-Israel-Stewart formalism, a matching between $\NFSYM$ results and the hydro expansion is only possible up to the first order, which gives rise to $η/s=1/4π$. Matching the results to the next order gives rise to causality/stability-violating values. Furthermore, we show that the pressure…
We propose a method which enables a QCD-based calculation of a relaxation time for a dissipative current in the causal and dissipative hydrodynamic equation derived by Israel and Stewart. We point out that the Israel-Stewart equation is not unique as a causal and dissipative hydrodynamic equation, and the form of the causal and dissipative hydrodynamic equation is determined by the shape of a spectral function reflecting the properties of elementary excitations in the system we consider. Our method utilizes a relaxation function, which can be calculated from QCD using the linear response theor…
Nonlinear Causality and Strong Hyperbolicity of Einstein-Israel-Stewart Theories of Transient Relativistic Fluid Dynamics
Department of Physics and Astronomy, University of Florence（佛罗伦萨大学物理与天文学系）；Department of Mathematics, Vanderbilt University（范德堡大学数学系）；Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学科学与技术学院）
Following the procedure introduced by Israel and Stewart, we expand the entropy current up to the third order in the shear stress tensor $π^{αβ}$ and derive a novel third-order evolution equation for $π^{αβ}$. This equation is solved for the one-dimensional Bjorken boost-invariant expansion. The scaling solutions for various values of the shear viscosity to the entropy density ratio $η/s$ are shown to be in very good agreement with those obtained from kinetic transport calculations. For the pressure isotropy starting with 1 at $τ_0=0.4 fm/c$, the third-order corrections to Israel-Stewart theor…
Tensors describing boost-invariant and cylindrically symmetric expansion of a relativistic dissipative fluid are decomposed in a suitable chosen basis of projection operators. This leads to a simple set of scalar equations which determine the fluid behavior. As special examples, we discuss the case of the Israel-Stewart theory and the model of highly-anisotropic and strongly-dissipative hydrodynamics ADHYDRO. We also introduce the matching conditions between the ADHYDRO description suitable for the very early stages of heavy-ion collisions and the Israel-Stewart theory applicable for later sta…
We study causal hydrodynamics (Israel-Stewart theory) of gauge theory plasmas from the AdS/CFT duality. Causal hydrodynamics requires new transport coefficients (relaxation times) and we compute them for a number of supersymmetric gauge theories including the N=4 SYM. However, the relaxation times obtained from the "shear mode" do not agree with the ones from the "sound mode," which implies that the Israel-Stewart theory is not a sufficient framework to describe the gauge theory plasmas.
We study the one-dimensional boost-invariant Boltzmann equation in the relaxation-time approximation using special moments of the distribution function for a system with a finite particle mass. The infinite hierarchy of moments can be truncated by keeping only the three lowest moments that correspond to the three independent components of the energy-momentum tensor. We show that such a three-moment truncation reproduces accurately the exact solution of the kinetic equation after a simple renormalization that takes into account the effects of the neglected higher moments. We derive second-order…
Israel-Stewart theory is a causal, stable formulation of relativistic dissipative fluid dynamics. This theory has been shown to give a decent description of the dynamical behavior of a relativistic fluid in cases where shear stress becomes important. In principle, it should also be applicable to situations where heat flow becomes important. However, it has been shown that there are cases where Israel-Stewart theory cannot reproduce phenomena associated with heat flow. In this paper, we derive a relativistic dissipative fluid-dynamical theory from kinetic theory which provides a good descriptio…
We investigate whether hydrodynamic attractors are present in simulations of the quark-gluon plasma formed in heavy-ion collisions. We argue that Lagrangian schemes to solve the relativistic viscous fluid equations can be particularly useful to characterize the properties of attractors in heavy ion collisions. Preliminary results are shown in Israel-Stewart theory to support such a claim. We also discuss how to perform the slow-roll expansion in Israel-Stewart theory undergoing a general flow, which naturally leads to the definition of a resummed shear viscosity coefficient that depends on the…
Focusing on the numerical aspects and accuracy we study a class of bulk viscosity driven expansion scenarios using the relativistic Navier-Stokes and truncated Israel-Stewart form of the equations of relativistic dissipative fluids in 1+1 dimensions. The numerical calculations of conservation and transport equations are performed using the numerical framework of flux corrected transport. We show that the results of the Israel-Stewart causal fluid dynamics are numerically much more stable and smoother than the results of the standard relativistic Navier-Stokes equations.
Based on the conservation-dissipation formalism proposed by Zhu and collaborators we formulate a general version of the Israel-Stewart theory for relativistic fluid dynamics with bulk viscosity. Our generalization consists in allowing for a wide range of dependence of the entropy density on the bulk viscosity. We show the existence of Godunov-Boillat variables for this model. By known properties of systems possessing such variables, this provides an alternative proof of the recently established existence of solutions for the Israel-Stewart theory locally in time, and a proof that entropy produ…
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.
We derive the equations of second order dissipative fluid dynamics from the relativistic Boltzmann equation following the method of W. Israel and J. M. Stewart. We present a frame independent calculation of all first- and second-order terms and their coefficients using a linearised collision integral. Therefore, we restore all terms that were previously neglected in the original papers of W. Israel and J. M. Stewart.
We derive the second order hydrodynamic equations for the relativistic system of multi-components with multiple conserved currents by generalizing the Israel-Stewart theory and Grad's moment method. We find that, in addition to the conventional moment equations, extra moment equations associated with conserved currents should be introduced to consistently match the number of equations with that of unknowns and to satisfy the Onsager reciprocal relations. Consistent expansion of the entropy current leads to constitutive equations which involve the terms not appearing in the original Israel-Stew…
We present a numerical code modeling the evolution of the medium formed in relativistic heavy ion collisions, ECHO-QGP. The code solves relativistic hydrodynamics in $(3+1)-$D, with dissipative terms included within the framework of Israel-Stewart theory; it can work both in Minkowskian and in Bjorken coordinates. Initial conditions are provided through an implementation of the Glauber model (both Optical and Monte Carlo), while freezeout and particle generation are based on the Cooper-Frye prescription. The code is validated against several test problems and shows remarkable stability and acc…
We review the traditional derivation of the fluid-dynamical equations from kinetic theory according to Israel and Stewart. We show that their procedure to close the fluid-dynamical equations of motion is not unique. Their approach contains two approximations, the first being the so-called 14-moment approximation to truncate the single-particle distribution function. The second consists in the choice of equations of motion for the dissipative currents. Israel and Stewart used the second moment of the Boltzmann equation, but this is not the only possible choice. In fact, there are infinitely man…
In the Israel-Stewart's 2nd order hydrodynamics, a viscous effect on dilepton emission from a QGP medium is investigated. Dileptons are strongly affected by QGP viscosity. Large invariant mass dileptons, due to their lower velocity, are less affected by viscosity than the low invariant mass dileptons. We also show that the ratio of photon to dilepton is sensitive to the viscosity and can serve as a viscometer for QGP.
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.
We investigate coefficients in the Israel-Stewart's causal hydrodynamics and discuss the way to calculate them with a microscopic theory. Based on the hadro-molecular simulation based on an event generator URASiMA, we evaluate the coefficients for a hot and dense hadronic fluid.
We present the results of deriving the Israel-Stewart equations of relativistic dissipative fluid dynamics from kinetic theory via Grad's 14-moment expansion. Working consistently to second order in the Knudsen number, these equations contain several new terms which are absent in previous treatments.
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.
In the Israel-Stewart's theory of second order hydrodynamics, we have analysed the recent PHENIX data on charged particles elliptic flow in Au+Au collisions. PHENIX data demand more viscous fluid in peripheral collisions than in central collisions. Over a broad range of collision centrality (0-10%- 50-60%), viscosity to entropy ratio ($η/s$) varies between 0-0.17.
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 $T_c$.
In the Israel-Stewart's theory of dissipative hydrodynamics, we have analysed the STAR data on $ϕ$ meson production in Au+Au collisions at $\sqrt{s}$=200 GeV. From a simultaneous fit to $ϕ$ mesons multiplicity, mean $p_T$ and integrated $v_2$, we obtain a phenomenological estimate of QGP viscosity, $η/s =0.07 \pm 0.03 \pm 0.14$, the first error is due to the experimental uncertainty in STAR measurements, the second reflects the uncertainties in initial and final conditions of the fluid.
In the framework of the Mueller-Israel-Stewart theory of dissipative fluid dynamics, we have studied the space-time evolution of a QGP fluid. For simplicity, we have considered shear viscosity only and neglected bulk viscosity and heat conductivity. Shear viscosity opposes the expansion and cooling of the fluid. As a result, the lifetime of the fluid is extended. We also find that the parton $p_T$ distribution is considerably flattened.
Viscous diffusion can broaden the rapidity dependence of two-particle transverse momentum fluctuations. Surprisingly, measurements at RHIC by the STAR collaboration demonstrate that this broadening is accompanied by the appearance of unanticipated structure in the rapidity distribution of these fluctuations in the most central collisions. Although a first order classical Navier-Stokes theory can roughly explain the rapidity broadening, it cannot explain the additional structure. We propose that the rapidity structure can be explained using the second order causal Israel-Stewart hydrodynamics w…
We investigate the microscopic origin of the relaxation time coefficient in relativistic fluid dynamics. We show that the extraction of the shear viscosity relaxation time via the gradient expansion is ambiguous and in general fails to give the correct result. The correct value for the shear viscosity relaxation time is extracted from the slowest non-hydrodynamic pole of the corresponding retarded Green's function, if such a pole is purely imaginary. According to the AdS/CFT correspondence, in strongly-coupled $\mathcal{N}=4$ SYM the non-hydrodynamic poles of the shear stress tensor nearest to…
We present the hybrid hadron string dynamic (HydHSD) model connecting the parton-hadron-string dynamic model (PHSD) and a hydrodynamic model taking into account shear viscosity within the Israel-Stewart approach. The performance of the code is tested on the pion and proton rapidity and transverse mass distributions calculated for Au+Au and Pb+Pb collision at AGS--SPS energies. The influence of the switch time from transport to hydro models, the viscous parameter, and freeze-out time are discussed. Since the applicability of the Israel-Stewart hydrodynamics assumes the perturbative character of…
In the Israel-Stewart's theory of 2nd order dissipative hydrodynamics, we have simulated $ϕ$ production in Au+Au and Cu+Cu collisions at $\sqrt{s}_{NN}$=200 GeV. Evolution of QGP fluid with viscosity over the entropy ratio $η/s$=0.25, thermalised at $τ_i$=0.2 fm, with initial energy density $ε_i$=5.1 $GeV/fm^3$ explains the experimental data on $ϕ$ multiplicity, integrated $v_2$, mean $p_T$, $p_T$ spectra and elliptic flow in central and mid-central Au+Au collisions. $η/s$=0.25 is also consistent with centrality dependence of $ϕ$ $p_T$ spectra in Cu+Cu collisions. The central energy density in…
Close to equilibrium, the kinetic coefficients of a thermodynamic system must satisfy a set of symmetry conditions, which follow from the Onsager-Casimir principle. Here, we show that, if a system of hydrodynamic equations is analysed from the perspective of the Onsager-Casimir principle, then it is possible to impose very strong symmetry conditions also on the principal part of such equations (the part with highest derivatives). In particular, we find that, in the absence of macroscopic magnetic fields and spins, relativistic hydrodynamics should always be symmetric-hyperbolic, when linearise…
