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 stages when the system is close to equilibrium.
It is shown that the boost-invariant and cylindrically non-symmetric hydrodynamic equations for baryon-free matter may be reduced to only two coupled differential equations. In the case where the system exhibits the cross-over phase transition, the standard numerical methods may be applied to solve these equations and the proposed scheme allows for a very convenient analysis of the cylindrically non-symmetric hydrodynamic expansion.
We show that the boost-invariant and cylindrically asymmetric hydrodynamic equations for baryon-free matter may be rewritten as only two coupled partial differential equations. In the case where the system exhibits the cross-over phase transition, the standard numerical methods may be applied to solve these equations. An example of our results describing non-central gold on gold collisions at RHIC energies is presented.
Simple, self-similar, analytic solutions of 1 + 1 dimensional relativistic hydrodynamics are presented, generalizing the Hwa - Bjorken boost-invariant solution to inhomogeneous rapidity distributions. These solutions are generalized also to 1 + 3 dimensional, cylindrically symmetric firetubes, corresponding to central collisions of heavy ions at relativistic bombarding energies.
We study the effects of hydrodynamic fluctuations in non-boost-invariant longitudinal expansion of matter formed in relativistic heavy ion collisions. We formulate the theory of thermal noise within second-order viscous hydrodynamics treating noise as a perturbation on top of the non-boost-invariant flow. We develop a numerical simulation model to treat the (1+1)-dimension hydrodynamic evolution. The code is tested to reproduce the analytic results for the Riemann solver for expansion of matter in vacuum. For viscous hydrodynamic expansion, the initial energy density distribution are obtained …
The recently formulated framework of anisotropic and dissipative hydrodynamics (ADHYDRO) is used to describe non-boost-invariant motion of the fluid created at the early stages of heavy-ion collisions. Very strong initial asymmetries of pressure are reduced by the entropy production processes. By the appropriate choice of the form of the entropy source we can reproduce realistic scenarios for the isotropization expected in heavy-ion collisions. Our previous results are generalized by including the realistic equation of state as the limit of the isotropization processes.
Explicit equations are given for describing the space-time evolution of non-ideal (viscous) relativistic fluids undergoing boost-invariant longitudinal and arbitrary transverse expansion. The equations are derived from the second-order Israel-Stewart approach which ensures causal evolution. Both azimuthally symmetric (1+1)-dimensional and non-symmetric (2+1)-dimensional transverse expansion are discussed. The latter provides the formal basis for the hydrodynamic computation of elliptic flow in relativistic heavy-ion collisions including dissipative effects.
The 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 formulate a general theory of thermal fluctuations within causal second-order viscous hydrodynamic evolution of matter formed in relativistic heavy-ion collisions. The fluctuation is treated perturbatively on top of a boost-invariant longitudinal expansion. Numerical simulation of thermal noise is performed for a lattice QCD equation of state and for various second-order dissipative evolution equations. Phenomenological effects of thermal fluctuations on the two-particle rapidity correlations are studied.
The Müller-Israel-Stewart second order theory of relativistic imperfect fluids based on Grad's moment method is used to study the expansion of hot matter produced in ultra-relativistic heavy ion collisions. The temperature evolution is investigated in the framework of the Bjorken boost-invariant scaling limit. The results of these second-order theories are compared to those of first-order theories due to Eckart and to Landau and Lifshitz and those of zeroth order (perfect fluid) due to Euler.
Relativistic non-ideal fluid dynamics is formulated in 3+1 space--time dimensions. The equations governing dissipative relativistic hydrodynamics are given in terms of the time and the 3-space quantities which correspond to those familiar from non-relativistic physics. Dissipation is accounted for by applying the causal theory of relativistic dissipative fluid dynamics. As a special case we consider a fluid without viscous/heat couplings in the causal system of transport/relaxation equations. For the study of physical systems we consider pure (1+1)-dimensional expansion in planar geometry, (1+…
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…
A recently formulated framework of highly-anisotropic and strongly-dissipative hydrodynamics (ADHYDRO) is used to describe the evolution of matter created in ultra-relativistic heavy-ion collisions. New developments of the model contain: the inclusion of asymmetric transverse expansion (combined with the longitudinal boost-invariant flow) and comparisons of the model results with the RHIC data, which have become possible after coupling of ADHYDRO with THERMINATOR. Various soft-hadronic observables (the transverse-momentum spectra, the elliptic flow coefficient v_2, and the HBT radii) are calcu…
We use a boost-invariant one-dimensional (cylindrically symmetric) fluid dynamics code to calculate $e^+e^-$ production from $ρ^0$ and $ω$ decay in the central rapidity region of a central S+Au collision at $\sqrt{s}=20$ GeV/nucleon. We use equations of state with a first-order phase transition between a massless pion gas and quark gluon plasma, with transition temperatures in the range $150-200$ MeV. The production cross section at the $ρ$ mass loosely constrains the transition and freeze-out temperatures, and we find that the $m_T$ spectrum is a good thermometer for sufficiently high $T_c$.
Non-equilibrium fluid dynamics derived from the extended irreversible thermodynamics of the causal Müller--Israel--Stewart theory of dissipative processes in relativistic fluids based on Grad's moment method is applied to the study of the dynamics of hot matter produced in ultra--relativistic heavy ion collisions. The temperature, energy density and entropy evolution are investigated in the framework of the Bjorken boost--invariant scaling limit. The results of these second order theories are compared to those of first order theories due to Eckart and to Landau and Lifshitz and those of zeroth…
The introduced earlier projection method for boost-invariant and cylindrically symmetric systems is used to introduce a new formulation of anisotropic hydrodynamics that allows for three substantially different values of pressure acting locally in three different directions. Our considerations are based on the Boltzmann kinetic equation with the collision term treated in the relaxation time approximation and the momentum anisotropy is included explicitly in the leading term of the distribution function. A novel feature of our work is the complete analysis of the second moment of the Boltzmann …
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 a boost-invariant one-dimensional (cylindrically symmetric) fluid dynamics code to calculate thermal photon production in the central rapidity region of S+Au and Pb+Pb collisions at SPS energy ($\sqrt{s}=20$ GeV/nucleon). We assume that the hot matter is in thermal equilibrium throughout the expansion, but consider deviations from chemical equilibrium in the high temperature (deconfined) phase. We use equations of state with a first-order phase transition between a massless pion gas and quark gluon plasma, with transition temperatures in the range $150 \leq T_c \leq 200$ MeV.
We study the dissipative evolution of (0+1)-dimensionally expanding media with Bjorken symmetry using the Boltzmann equation for massive particles in relaxation-time approximation. Breaking conformal symmetry by a mass induces a non-zero bulk viscous pressure in the medium. It is shown that even a small mass (in units of the local temperature) drastically modifies the well-known attractor for the shear Reynolds number previously observed in massless systems. For generic nonzero particle mass, neither the shear nor the bulk viscous pressure relax quickly to a non-equilibrium attractor; they app…
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…
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…
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…
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…
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…
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…
Dissipative spin hydrodynamics in Bjorken flow and thermal dilepton production
Birla Institute of Technology and Science Pilani（比拉理工学院皮拉尼分校）；National Institute of Science Education and Research（国家科学教育与研究机构）；Institute of Physics（物理研究所）
We have studied the effect of nonzero bulk viscosity with peak near the lattice QCD predicted crossover temperature $T_{co}\sim 175 MeV$ on charged particle transverse momentum spectra and elliptic flow. The Israel-Stewart theory of 2nd order causal dissipative relativistic fluid dynamics is used to simulate the space time evolution of the matter formed in Au-Au collisions at $\sqrt{s_{NN}}$=200 GeV assuming longitudinal boost invariance. A systematic comparison of temperature, transverse velocity, spatial and momentum anisotropy evolution of the ideal, bulk and shear viscous fluid has been ca…
The first order hydrodynamic evolution equations for the shear stress tensor, the bulk viscous pressure and the charge current have been studied for a system of quarks and gluons, with a non-vanishing quark chemical potential and finite quark mass. The first order transport coefficients have been obtained by solving an effective Boltzmann equation for the grand-canonical ensemble of quasiquarks and quasigluons. We adopted temperature dependent effective fugacity for the quasiparticles to encode the hot QCD medium effects. The non-trivial energy dispersion of the quasiparticles induces mean fie…
We present a dynamical scheme for hadronization with first-order confinement phase transition. The thermodynamical conditions of phase equilibrium, the fluid velocity profile, and the dissipative effect determine the macroscopic changes of the parton volume and the corresponding hadron volume during the phase transition. The macroscopic volume changes are the basis for building up a dynamical scheme by considering microscopic transition processes from partons to hadrons and backwards. The established scheme is proved by comparing the numerical results with the analytical solutions in the case …
[Background] Experimental data from heavy-ion experiments at RHIC-BNL and LHC-CERN are quantitatively described using relativistic fluid dynamics. Even p+A and p+p collisions show signs of collective behavior describable in the same manner. Nevertheless, small system sizes and large gradients strain the limits of applicability of fluid-dynamical methods. [Purpose] The range of applicability of fluid dynamics for the description of the collective behavior, and in particular of the elliptic flow, of small systems needs to be explored. [Method] Results of relativistic fluid-dynamical simulations …
Using the AdS/CFT correspondence, we compute analytically the late-time behavior of two-point functions, Wilson loops and entanglement entropy in a strongly coupled $\mathcal{N}=4$ super-Yang-Mills plasma undergoing a boost-invariant expansion. We take into account the effects of first order dissipative hydrodynamics and investigate the effects of the (time-dependent) shear viscosity on the various observables. The two-point functions decay exponentially at late times and are unaffected by the viscosity if the points are separated along the transverse directions. For longitudinal separation we…
General equilibrium second-order hydrodynamic coefficients for free quantum fields
Universitá di Firenze（佛罗伦萨大学）；INFN Sezione di Firenze（意大利国家核物理研究所佛罗伦萨分部）；Institut für Theoretische Physik, Universität Heidelberg（海德堡大学理论物理研究所）；New York University Florence（纽约大学佛罗伦萨分校）
Relativistic Covariance from Hamiltonian Hydrodynamics in the Chiral Limit
School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）
We derive the equations of motion for a system undergoing boost-invariant longitudinal and azimuthally-symmetric transverse "Gubser flow" using leading-order anisotropic hydrodynamics. This is accomplished by assuming that the one-particle distribution function is ellipsoidally-symmetric in the momenta conjugate to the de Sitter coordinates used to parameterize the Gubser flow. We then demonstrate that the SO(3)_q symmetry in de Sitter space further constrains the anisotropy tensor to be of spheroidal form. The resulting system of two coupled ordinary differential equations for the de Sitter-s…
A recently developed framework of highly-anisotropic and strongly-dissipative hydrodynamics -- ADHYDRO -- has been introduced and used to analyze the space-time evolution of matter produced in ultra-relativistic heavy-ion collisions. The main goal of this analysis was to study the effect of initial highly-anisotropic stages on the final soft hadronic observables typically measured in the experiment. The study was done in the context of the heavy-ion measurements performed at RHIC (Relativistic Heavy Ion Collider) in Brookhaven National Laboratory. Starting from the general assumption about the…
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 …
