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 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.
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…
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…
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 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 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…
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…
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…
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 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…
We review the entropy based derivation of third-order hydrodynamic equations and compare their solutions in one-dimensional boost-invariant geometry with calculations by the partonic cascade BAMPS. We demonstrate that Grad's approximation, which underlies the derivation of both Israel-Stewart and third-order equations, describes the transverse spectra from BAMPS with high accuracy. At the same time solutions of third-order equations are much closer to BAMPS results than solutions of Israel-Stewart equations. Introducing a resummation scheme for all higher-oder corrections to one-dimensional hy…
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…
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.
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）
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.
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 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…
Based on the exact solution of Boltzmann kinetic equation in the relaxation-time approximation, the precision of the two most recent formulations of relativistic second-order non-conformal viscous hydrodynamics (14-moment approximation and causal Chapman-Enskog method), standard Israel-Stewart theory, and anisotropic hydrodynamics framework, in the simple case of one-dimensional Bjorken expansion, is tested. It is demonstrated that the failure of Israel-Stewart theory in reproducing exact solutions of the Boltzmann kinetic equation occurs due to neglecting and/or choosing wrong forms of some o…
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.
We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a 0+1d conformal system using kinetic theory in the relaxation time approximation. We compare our results to the non-equilibrium attractor obtained from exact solution of the 0+1d conformal Boltzmann equation, Navier-Stokes theory, and second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal to transverse…
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…
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 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…
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.
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.
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.
Based on the Nakajima-Zubarev type nonequilibrium density operator, we derive a hyperbolic hydrodynamical equation. Microscopic Kubo-formulas for all coefficients in the hyperbolic hydrodynamics are obtained. Coefficients $α_{i}$'s and $β_{i}$'s in the Israel-Stewart equation are given as current-weighted correlation lengths which are to be calculated in statistical mechanics.
In this paper, we develop a stable and fast numerical scheme for relativistic dissipative hydrodynamics based on Israel-Stewart theory. Israel-Stewart theory is a stable and causal description of dissipation in relativistic hydrodynamics although it includes relaxation process with the timescale for collision of constituent particles, which introduces stiff equations and makes practical numerical calculation difficult. In our new scheme, we use Strang's splitting method, and use the piecewise exact solutions for solving the extremely short timescale problem. In addition, since we split the cal…
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…
There is growing evidence that the hydrodynamic gradient expansion is factorially divergent. We advocate for using Dingle's singulants as a way to gain analytic control over its large-order behaviour for nonlinear flows. Within our approach, singulants can be viewed as new emergent degrees of freedom which reorganise the large-order gradient expansion. We work out the physics of singulants for longitudinal flows, where they obey simple evolution equations which we compute in Müller-Israel-Stewart-like models, holography and kinetic theory. These equations determine the dynamics of the large-or…
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…
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.
In this work, we perform a phenomenological derivation of the first- and second-order relativistic hydrodynamics of dissipative fluids. To set the stage, we start with a review of the ideal relativistic hydrodynamics from energy-momentum and particle number conservation equations. We then go on to discuss the matching conditions to local thermodynamical equilibrium, symmetries of the energy-momentum tensor, decomposition of dissipative processes according to their Lorentz structure, and finally, the definition of the fluid velocity in the Landau and Eckart frames. With this preparatory work, w…
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.
In this paper we present a method to improve the description of 0+1 dimensional boost invariant dissipative dynamics in the presence of large momentum-space anisotropies. We do this by reorganizing the canonical hydrodynamic expansion of the distribution function around a momentum-space anisotropic ansatz rather than an isotropic equilibrium one. At leading order the result obtained is two coupled ordinary differential equations for the momentum-space anisotropy and typical momentum of the degrees of freedom. We show that this framework can reproduce both the ideal hydrodynamic and free stream…
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 $T_c$.
