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-Stewart theory even in the single component limit. We also find several terms which exhibit thermal diffusion such as Soret and Dufour effects. We finally compare our results with those of other existing formalisms.
Dissipative processes in relativistic fluids are known to be important in the analyses of the hot QCD matter created in high-energy heavy-ion collisions. In this work, I consider dissipative corrections to energy and conserved charge densities, which are conventionally assumed to be vanishing but could be finite. Causal dissipative hydrodynamics is formulated in the presence of those dissipative currents. The relation between hydrodynamic stability and transport coefficients is discussed. I then study their phenomenological consequences on the observables of heavy-ion collisions in numerical s…
I present an overview of recent theoretical results on fluctuations of conserved charges in heavy-ion collisions obtained in relativistic hydrodynamics and molecular dynamics frameworks. In particular, I discuss the constraints on the location of the QCD critical point based on comparisons of experimental data on proton number cumulants with precision calculations of non-critical contributions. Recent developments on critical fluctuations in molecular dynamics simulations are covered as well.
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.
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 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 introduce a fast and simple method of computing cumulants of net-proton or net-charge fluctuations in event-by-event hydrodynamic simulations of heavy-ion collisions. One evaluates the mean numbers of particles in every hydrodynamic event. Cumulants are then expressed as a function of these mean numbers. We implement the corrections due to global conservation laws. The method is tested using ideal hydrodynamic simulations of Au+Au collisions at $\sqrt{s_{NN}}=200$ AGeV with the NeXSPheRIO code. Results are in good agreement with experimental data on net-proton and net-charge fluctuations by…
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.
The quark-gluon plasma behaves as a relativistic viscous fluid in high-energy heavy ion collisions. I develop a causal dissipative hydrodynamic model at finite baryon density for RHIC and LHC to estimate the net baryon rapidity distribution. The net baryon number is found to be carried to forward rapidity by the flow, effectively enhancing the transparency of the collisions. This suggests that the energy available for the production of a hot medium could be larger than that naively implied from experimental data. Also the distribution can be sensitive to baryon dissipation as much as to shear …
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.
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.
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…
Recent discussions of RHIC data emphasized the exciting possibility that the matter produced in nucleus-nucleus collisions shows properties of a near-perfect fluid. Here, we aim at delineating the applicability of fluid dynamics, which is needed to quantify the size of corresponding dissipative effects. We start from the equations for dissipative fluid dynamics, which we derive from kinetic theory up to second order (Israel-Stewart theory) in a systematic gradient expansion. In model studies, we then establish that for too early initialization of the hydrodynamic evolution ($τ_0 \lsim 1$ fm/c)…
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 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 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…
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.
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.
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.
Relativistic thermodynamics is derived from kinetic equilibrium in a general frame. Based on a novel interpretation of Lagrange multipliers in the equilibrium state we obtain a generic stable but first order relativistic dissipative hydrodynamics. Although this was believed to be impossible, we circumvent this difficulty by a specific handling of the heat flow.
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…
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…
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…
In this review, we present the motivation for using relativistic anisotropic hydrodynamics to study the physics of ultrarelativistic heavy-ion collisions. We then highlight the main ingredients of the 3+1D quasiparticle anisotropic hydrodynamics model and present phenomenological comparisons with experimental data at different collision energies. These comparisons show that anisotropic hydrodynamics can describe many bulk observables of the quark-gluon plasma.
We have computed the circulation integrations of thermal vorticity with and without charged currents in dissipative fluids. We find that the relativistic Kelvin circulation theorem will be modified by the dissipative effects, therefore, the circulation integrations of thermal vorticity may not be conserved during the fluid evolution.
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 briefly discuss the recent developments in causal dissipative hydrodynamic for relativistic heavy ion collisions. Phenomenological estimate of QGP viscosity over entropy ratio from several experimental data, e.g. STAR's $ϕ$ meson data, centrality dependence of elliptic flow, universal scaling elliptic flow etc. are discussed. QGP viscosity, extracted from hydrodynamical model analysis can have very large systematic uncertainty due to uncertain initial conditions.
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 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 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…
Causality and Stability of First-Order Relativistic Spin Hydrodynamics with Conserved Charges
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（三峡大学天文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）
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.
Development of a new framework for derivation of order-by-order hydrodynamics from Boltzmann equation is necessary as the widely used Anderson-Witting formalism leads to violation of fundamental conservation laws when the relaxation-time depends on particle energy, or in a hydrodynamic frame other than the Landau frame. We generalize an existing framework for consistent derivation of relativistic dissipative hydrodynamics from the Boltzmann equation with a energy-dependent relaxation-time by extending the Anderson-Witting relaxation-time approximation. We argue that the present framework is co…
We derive relativistic dissipative spin hydrodynamics from kinetic theory featuring a nonlocal collision term using the method of moments. In this framework, the components of the spin tensor are dynamical variables which obey relaxation-type equations. We find that the corresponding relaxation times are determined by the local part of the collision term, while the nonlocal part contributes to the Navier-Stokes terms in these equations of motion. The spin relaxation time scales are comparable to those of the usual dissipative currents. Finally, the Navier-Stokes limit of the Pauli-Lubanski vec…
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…
We extended our formulation of causal dissipative hydrodynamics [T. Koide \textit{et al.}, Phys. Rev. \textbf{C75}, 034909 (2007)] to be applicable to the ultra-relativistic regime by considering the extensiveness of irreversible currents. The new equation has a non-linear term which suppresses the effect of viscosity. We found that such a term is necessary to guarantee the positive definiteness of the inertia term and stabilize numerical calculations in ultra-relativistic initial conditions. Because of the suppression of the viscosity, the behavior of the fluid is more close to that of the id…
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…
Starting from the Boltzmann equation in the relaxation time approximation and employing a Chapman-Enskog like expansion for the distribution function close to equilibrium, we derive second-order evolution equations for the shear stress tensor and the dissipative charge current for a system of massless quarks and gluons. The transport coefficients are obtained exactly using quantum statistics for the phase space distribution functions at non-zero chemical potential. We show that, within the relaxation time approximation, the second-order evolution equations for the shear stress tensor and the d…
Recently we proposed a novel approach to the formulation of relativistic dissipative hydrodynamics by extending the so-called matching conditions in the Eckart frame [Phys. Rev. {\bf C 85}, (2012) 14906]. We extend this formalism further to the arbitrary Lorentz frame. We discuss the stability and causality of solutions of fluid equations which are obtained by applying this formulation to the Landau frame, which is more relevant to treat the fluid produced in ultra-relativistic heavy-ion collisions. We derive equations of motion for a relativistic dissipative fluid with zero baryon chemical po…
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…
In a first order theory of dissipative hydrodynamics, we have simulated hydrodynamic evolution of QGP fluid with dissipation due to shear viscosity only. Simulation confirms that compared to an ideal fluid, energy density or temperature of a viscous fluid evolve slowly. Transverse expansion is also more in viscous fluid. We also study the effect of viscosity on particle production. Particle production is enhanced, more at large $p_T$. The elliptic flow on the otherhand decreases and shows a tendency to saturate at large $p_T$.
We show that the relativistic dissipative hydrodynamic equation derived from the relativistic Boltzmann equation by the renormalization-group method uniquely leads to the one in the energy frame proposed by Landau and Lifshitz, provided that the macroscopic-frame vector, which defines the local rest frame of the fluid velocity, is independent of the momenta of constituent particles, as it should. We argue that the relativistic hydrodynamic equations for viscous fluids must be defined on the energy frame if it is consistent with the underlying relativistic kinetic equation.
We compute dilepton production from the deconfined phase of the quark-gluon plasma using leading-order (3+1)-dimensional anisotropic hydrodynamics. The anisotropic hydrodynamics equa- tions employed describe the full spatiotemporal evolution of the transverse temperature, spheroidal momentum-space anisotropy parameter, and the associated three-dimensional collective flow of the matter. The momentum-space anisotropy is also taken into account in the computation of the dilepton production rate, allowing for a self-consistent description of dilepton production from the quark-gluon plasma. For our…
