The stability and causality of the Landau-Lifshitz theory and the Israel-Stewart type causal dissipative hydrodynamics are discussed. We show that the problem of acausality and instability are correlated in relativistic dissipative hydrodynamics and instability is induced by acausality. We further discuss the stability of the scaling solution. The scaling solution of the causal dissipative hydrodynamics can be unstable against inhomogeneous perturbations.
We investigate the causality and stability of relativistic dissipative fluid dynamics in the absence of conserved charges. We perform a linear stability analysis in the rest frame of the fluid and find that the equations of relativistic dissipative fluid dynamics are always stable. We then perform a linear stability analysis in a Lorentz-boosted frame. Provided that the ratio of the relaxation time for the shear stress tensor, $τ_π$, to the sound attenuation length, $Γ_s = 4η/3(\varepsilon+P)$, fulfills a certain asymptotic causality condition, the equations of motion give rise to stable solut…
We propose a stable first-order relativistic dissipative hydrodynamic equation in the particle frame (Eckart frame) for the first time. The equation to be proposed was in fact previously derived by the authors and a collaborator from the relativistic Boltzmann equation. We demonstrate that the equilibrium state is stable with respect to the time evolution described by our hydrodynamic equation in the particle frame. Our equation may be a proper starting point for constructing second-order causal relativistic hydrodynamics, to replace Eckart's particle-flow theory.
We argue that there is correspondence between the perfect nonextensive hydrodynamics and the usual dissipative hydrodynamics, which we call nonextensive/dissipative correspondence (NexDC). It leads to simple expression for dissipative entropy current and allows for predictions for the ratio of bulk and shear viscosities to entropy density, $ζ/s$ and $η/s$.
We argue that there is profound correspondence (the nonextensive/dissipative correspondence - NexDC) between the perfect nonextensive hydrodynamics and the usual dissipative hydrodynamics which leads to simple expression for dissipative entropy current.
We demonstrate that nonextensive perfect relativistic hydrodynamics ($q$-hydrodynamics) can serve as a model of the usual relativistic dissipative hydrodynamics ($d$-hydrodynamics) facilitating therefore considerably its applications. As illustration we show how using $q$-hydrodynamics one gets the $q$-dependent expressions for the dissipative entropy current and the corresponding ratios of the bulk and shear viscosities to entropy density, $ζ/s$ and $η/s$.
We derive the constitutive equations of causal relativistic dissipative hydrodynamics ($d$-hydrodynamics) from perfect nonextensive hydrodynamics ($q$-hydrodynamics) using the nonextensive/dissipative correspondence (NexDC) proposed by us recently. The $q$-hydrodynamics can be thus regarded as a possible model for the $d$-hydrodynamics facilitating its application to high energy multiparticle production processes. As an example we have shown that applying the NexDC to the perfect 1+1 $q$-hydrodynamics, one obtains a proper time evolution of the bulk pressure and the Reynolds number.
We propose to model the dissipative hydrodynamics used in description of the multiparticle production processes ($d$-hydrodynamics) by a special kind of the perfect nonextensive fluid ($q$-fluid) where $q$ denotes the nonextensivity parameter appearing in the nonextensive Tsallis statistics. The advantage of $q$-hydrodynamics lies in its formal simplicity in comparison to the $d$-hydrodynamics. We argue that parameter $q$ describes summarily (at least to some extent) all dynamical effects behind the viscous behavior of the hadronic fluid.
We propose an effective action for first order relativistic dissipative hydrodynamics that can be used to evaluate n-point symmetrized correlation functions, taking into account thermal fluctuations of the hydrodynamic variables.
We study the causality and stability of relativistic hydrodynamics with the inclusion of the spin degree of freedom as a hydrodynamic field. We consider two specific models of spin-hydrodynamics for this purpose. A linear mode analysis for static background shows that a first-order dissipative spin-hydrodynamics remains acausal and admits instabilities. Besides, it is found that the inclusion of the spin field in hydrodynamics leads to new kinds of linear modes in the system. These new modes also exhibit instability and acausal behavior. The second model of the spin-hydrodynamics that we have …
Kinetic and hydrodynamic models describing early stages of relativistic heavy-ion collisions are discussed. We emphasise the role of the shear-bulk coupling for the correct determination of the time dependence of the bulk viscous pressure.
High-energy heavy ion collider experiments at RHIC and LHC have revealed that relativistic hydrodynamic models describe the hot and dense quark matter quantitatively. In this study, I develop a novel dissipative hydrodynamic model at finite baryon density to investigate the net baryon rapidity distribution. The results show that the distribution is widened in hydrodynamic evolution, which implies that the transparency of the collisions is effectively enhanced. This suggests that the kinetic energy loss for medium production at the initial stage could be larger. Furthermore, the net baryon dist…
We argue that different formulations of hydrodynamics are related to uncertainties in the definitions of local thermodynamic and hydrodynamic variables. We show that this ambiguity can be resolved by viewing different formulations of hydrodynamics as particular "gauge choices" which lead to the same physical behavior of the system. Using the example of bulk viscosity, we show that Bemfica-Disconzi-Noronha-Kovtun (BDNK) and Israel-Stewart hydrodynamics are particular "gauge choices" of this type, related by a well-defined transformation of thermodynamic and hydrodynamic variables. We argue that…
In this contribution we study the effects of three transport coefficients of dissipative hydrodynamics on thermal dilepton anisotropic flow observables. The first two transport coefficients investigated influence the overall size and growth rate of shear viscous pressure, while the last transport coefficient governs the magnitude of net baryon number diffusion in relativistic dissipative fluid dynamics. All calculations are done using state-of-the-art 3+1D hydrodynamical simulations. We show that thermal dileptons are sensitive probes of the transport coefficients of dissipative hydrodynamics.
We studied the shock propagation and its stability with the causal dissipative hydrodynamics in 1+1 dimensional systems. We show that the presence of the usual viscosity is not enough to stabilize the solution. This problem is solved by introducing an additional viscosity which is related to the coarse-graining scale of the theory.
Relativistic dissipative hydrodynamics including hydrodynamic fluctuations is formulated by putting an emphasis on non-linearity and causality. As a consequence of causality, dissipative currents become dynamical variables and noises appeared in an integral form of constitutive equations should be colored ones from fluctuation-dissipation relations. Nevertheless noises turn out to be white ones in its differential form when noises are assumed to be Gaussian. The obtained ifferential equations are very useful in numerical implementation of relativistic fluctuating hydrodynamics.
We have derived the analytic solutions of dissipative relativistic spin hydrodynamics with Gubser expansion. Following the standard strategy of deriving the solutions in a Gubser flow, we take the Weyl rescaling and obtain the energy-momentum and angular momentum conversation equations in the $dS_{3}\times\mathbb{R}$ space-time. We then derive the analytic solutions of spin density, spin potential and other thermodynamic in $dS_{3}\times\mathbb{R}$ space-time and transform them back into Minkowski space-time $\mathbb{R}^{3,1}$. In the Minkowski space-time, the spin density and spin potential i…
The microscopic formulae of the bulk viscosity $ζ$ and the corresponding relaxation time $τ_Π$ in causal dissipative relativistic fluid dynamics are derived by using the projection operator method. In applying these formulae to the pionic fluid, we find that the renormalizable energy-momentum tensor should be employed to obtain consistent results. In the leading order approximation in the chiral perturbation theory, the relaxation time is enhanced near the QCD phase transition and $τ_Π$ and $ζ$ are related as $τ_Π=ζ/[β\{(1/3-c_{s}^{2})(ε+P)-2(ε-3P)/9\}]$, where $ε$, $P$ and $c_{s}$ are the ene…
The quark-gluon plasma is considered to behave as a relativistic viscous fluid in the high-energy heavy ion collisions. In this study, I develop and estimate a second order dissipative hydrodynamic model at finite baryon density with effects of baryon dissipation together with those of shear and bulk viscosities. It is found that the hydrodynamic evolution effectively reduces baryon stopping, suggesting that the collisions are less transparent at the initial stage. Also the net baryon distribution is found sensitive to baryon dissipation as well as to viscosities. The results indicate that the…
We derive the multi-component second-order dissipative relativistic hydrodynamic equations using the moment-expansion method. By computing the transport coefficients using hard-sphere interactions, we investigate the role of multiple components and the reactive collisions. We find that both of these factors increase the effective cross-section and hence decrease the transport coefficients and relaxation times. We further compute such transport properties using leading-order perturbative QCD cross-sections. For both types of cross-sections, we find that the ratio between vector current relaxati…
Relativistic dissipative hydrodynamic equations are extended by taking into account particle number changing processes in a gluon system, which expands in one dimension boost-invariantly. Chemical equilibration is treated by a rate equation for the particle number density based on Boltzmann equation and Grad's ansatz for the off-equilibrium particle phase space distribution. We find that not only the particle production, but also the temperature and the momentum spectra of the gluon system, obtained from the hydrodynamic calculations, are sensitive to the rates of particle number changing proc…
The quark gluon plasma generated in ultrarelativistic heavy ion collisions may possess sizable momentum-space anisotropies that cause the longitudinal and transverse pressures in the local rest frame to be significantly different. We review recent attempts to derive a dynamical framework that can reliably describe systems that possess a high degree of momentum-space anisotropy. The dynamical framework that has been developed can describe the evolution of the quark gluon plasma ranging from the longitudinal free-streaming limit to the ideal hydrodynamical limit.
Starting from Boltzmann equation with relaxation time approximation for the collision term and using Chapman-Enskog like expansion for distribution function close to equilibrium, we derive hydrodynamic evolution equations for the dissipative quantities directly from their definition. Although the form of the equations is identical to those obtained in traditional Israel-Stewart approaches employing Grad's 14-moment approximation and second moment of Boltzmann equation, the coefficients obtained are different. In the case of one-dimensional scaling expansion, we demonstrate that our results are…
We explore theoretical uncertainties in the hydrodynamic description of relativistic heavy-ion collisions by examining the full non-linear causality conditions and quantifying the second-order transport coefficients' role on flow observables. The causality conditions impose physical constraints on the maximum allowed values of inverse Reynolds numbers during the hydrodynamic evolution. Including additional second-order gradient terms in the Denicol-Niemi-Molnár-Rischke (DNMR) theory significantly shrinks the casual regions compared to those in the Israel-Stewart hydrodynamics. For Au+Au collis…
Second-order dissipative hydrodynamic equations for each component of a multi-component system are derived using the entropy principle. The shear viscosity of the whole system, appearing in the equation summed-up over all components, is related to the partial shear pressures and cannot be considered as an external parameter. We demonstrate that it is essential to solve hydrodynamic equations for each component, instead of treating a mixture as an effective one-component system with a free parameter $η/s$. Thus, extractions of the $η/s$ value of the QGP at RHIC and LHC have to be reexamined.
We present a new formalism for the theory of relativistic dissipative hydrodynamics. Here, we look for the minimal structure of such a theory which satisfies the covariance and causality by introducing the memory effect in irreversible currents. Our theory has a much simpler structure and thus has several advantages for practical purposes compared to the Israel-Stewart theory (IS). It can readily be applied to the full three-dimensional hydrodynamical calculations. We apply our formalism to the Bjorken model and the results are shown to be analogous to the IS.
In this paper we consider dissipative hydrodynamic equations for systems with continuous broken symmetries. We first present the case of superfluidity, in which the symmetry U(1) is broken and then generalize to the chiral symmetry $SU(2)_L \times SU(2)_R$. The corresponding new transport coefficients are introduced.
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 aniso…
We apply divergence-type theory (DTT) dissipative hydrodynamics to study the 2+1 space-time evolution of the fireball created in Au+Au relativistic heavy-ion collisions at $\sqrt{s_{NN}}=$200 GeV. DTTs are exact hydrodynamic theories that do no rely on velocity gradient expansions and therefore go beyond second-order theories. We numerically solve the equations of motion of the DTT for Glauber initial conditions and compare the results with those of second-order theory based on conformal invariants (BRSS) and with data. We find that the charged-hadron minumum-bias elliptic flow reaches its max…
The first 3D calculation of shock wave propagation in a homogeneous QGP has been performed within the new formulation of relativistic dissipative hydrodynamics which preserves the causality. We found that the relaxation time plays an important role and also affects the angle of Mach cone.
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…
The stability conditions of a relativistic hydrodynamic theory can be derived directly from the requirement that the entropy should be maximised in equilibrium. Here we use a simple geometrical argument to prove that, if the hydrodynamic theory is stable according to this entropic criterion, then localised perturbations to the equilibrium state cannot propagate outside their future light-cone. In other words, within relativistic hydrodynamics, acausal theories must be thermodynamically unstable, at least close to equilibrium. We show that the physical origin of this deep connection between sta…
We present a new derivation of relativistic second-order dissipative hydrodynamics for quantum systems using Zubarev's non-equilibrium statistical-operator formalism. This is achieved by a systematic expansion of the energy-momentum tensor and the charge current to second order in deviations from equilibrium. As a concrete example, we obtain the relaxation equations for the shear-stress tensor, the bulk-viscous pressure, and the charge-diffusion currents required to close the set of equations of motion for relativistic second-order dissipative hydrodynamics. We also identify new transport coef…
Causality and stability analysis of relativistic spin hydrodynamics: Insights from a nonvanishing spin density background
College of Mathematics and Physics, China Three Gorges University（三峡大学数学与物理学院）；Center for Astronomy and Space Sciences and Institute of Modern Physics, China Three Gorges University（三峡大学天体物理与空间科学中心及现代物理研究所）；Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University（华中师范大学夸克与轻子物理教育部重点实验室及粒子物理研究所）
Using second--order dissipative hydrodynamics coupled self-consistently to the linear $σ$ model we study the 2+1 dimensional evolution of the fireball created in Au+Au relativistic collisions. We analyze the influence of the dynamics of the chiral fields on the charged-hadron elliptic flow $v_2$ and on the ratio $v_4/(v_2)^2$ for a temperature-independent as well as for a temperature-dependent viscosity-to-entropy ratio $η/s$ calculated from the linearized Boltzmann equation in the relaxation time approximation. We find that $v_2$ is not very sensitive to the coupling of chiral sources to the …
We have studied analytically the longitudinally boost-invariant motion of a relativistic dissipative fluid with spin. We have derived the analytic solutions of spin density and spin chemical potential as a function of proper time $τ$ in the presence of viscous tensor and the second order relaxation time corrections for spin. Interestingly, analogous to the ordinary particle number density and chemical potential, we find that the spin density and spin chemical potential decay as $\simτ^{-1}$ and $\simτ^{-1/3}$, respectively. It implies that the initial spin density may not survive at the freeze…
We use the entropy production variational method to associate a one particle distribution function to the assumed known energy-momentum and entropy currents describing a relativistic conformal fluid. Assuming a simple form for the collision operator we find this one particle distribution function explicitly, and show that this method of linking the hydro and kinetic description is a non trivial generalization of Grad's ansatz. The resulting constitutive relations are the same as in the conformal dissipative type theories discussed in J. Peralta-Ramos and E. Calzetta, Phys. Rev. D {\bfseries 80…
We deal with a novel approach to formulation of the relativistic dissipative hydrodynamics by extending the so-called matching conditions widely used in the literature. The form of the non-equilibrium entropy current can be determined by requiring thermodynamical stability of the entropy current under extended matching conditions. We derive equations of motion for the relativistic dissipative fluid based on the Eckart theory and show that linearized equations obtained from them are stable against small perturbations. It is also shown that the required fluid stability conditions are related to …
We present a new derivation of relativistic dissipative hydrodynamic equations, which invokes the second law of thermodynamics for the entropy four-current expressed in terms of the single-particle phase-space distribution function obtained from Grad's 14-moment approximation. This derivation is complete in the sense that all the second-order transport coefficients are uniquely determined within a single theoretical framework. In particular, this removes the long-standing ambiguity in the relaxation time for bulk viscosity thereby eliminating one of the uncertainties in the extraction of the s…
We introduce a new framework of highly-anisotropic hydrodynamics that includes dissipation effects. Dissipation is defined by the form of the entropy source that depends on the pressure anisotropy and vanishes for the isotropic fluid. With a simple ansatz for the entropy source obeying general physical requirements, we are led to a non-linear equation describing the time evolution of the anisotropy in purely-longitudinal boost-invariant systems. Matter that is initially highly anisotropic approaches naturally the regime of the perfect fluid. Thus, the resulting evolution agrees with the expect…
The recently formulated model of highly-anisotropic and strongly dissipative hydrodynamics is used in 3+1 dimensions to describe flow characteristics and strangeness production in Au+Au collisions at the highest RHIC beam energy. Our results show very weak dependence on the initial momentum anisotropy, provided the anisotropic phase lasts no longer than 1 fm/c.
With the goal of deriving dissipative hydrodynamics from an action, we study classical actions for open systems, which follow from the generic structure of effective actions in the Schwinger-Keldysh Closed-Time-Path formalism with two time axes and a doubling of degrees of freedom. The central structural feature of such effective actions is the coupling between degrees of freedom on the two time axes. This reflects the fact that from an effective field theory point of view, dissipation is the loss of energy of the low-energy hydrodynamical degrees of freedom to the integrated-out, UV degrees o…
