New, analytic solutions of relativistic viscous hydrodynamics are presented, describing expanding fireballs with Hubble-like velocity profile and ellipsoidal symmetry, similar to fireballs created in heavy ion collisions. We find that with these specifications, one obtains solutions where the shear viscosity essentially does not influence the time evolution of the system, thus these solutions are particularly adept tools to study the effect of bulk viscosity alone, which always results in a slower decrease of energy density as well as temperature compared to the case of perfect fluid. We investigate different scenarios for the bulk viscosity and find qualitatively different effects on the time evolution which suggests that there is a possibility to infer the value of bulk viscosity from energy density and temperature measurements in high-energy heavy-ion collisions.
A new family of simple, analytic solutions of self-similarly expanding fireballs is found for systems with ellipsoidal symmetry and a direction dependent, generalized Hubble flow. Gaussian, shell like or oscillating density profiles emerge for simple choices of an arbitrary scaling function. New, cylindrically or spherically symmetric as well as approximately one dimensional hydrodynamical solutions are obtained for various special choices of the initial conditions.
The ellipsoidally symmetric extension of Buda-Lund hydrodynamic model is shown here to yield a natural description of the pseudorapidity dependence of the elliptic flow $v_2(η)$, as determined recently by the PHOBOS experiment for Au+Au collisions at $\sqrt{s_{NN}} = 130$ and 200 GeV. With the same set of parameters, the Buda-Lund model describes also the transverse momentum dependence of $v_2$ of identified particles at mid-rapidity. The results confirm the indication for quark deconfinement in Au+Au collisions at RHIC, obtained from a successful Buda-Lund hydro model fit to the single partic…
Simple, self-similar, analytic solutions of (1+3)-dimensional relativistic hydrodynamics are presented for ellipsoidally symmetric finite fireballs corresponding to non-central collisions of heavy ions at relativistic bombarding energies. The hydrodynamical solutions are obtained for a new, general family of equations of state with the possibility of describing phase transitions.
A new class of analytic, exact, rotating, self-similar and surprisingly simple solutions of non-relativistic hydrodynamics are presented for a three-dimensionally expanding, spheroidally symmetric fireball. These results generalize earlier, non-rotating solutions for ellipsoidally symmetric fireballs with directional, three-dimensional Hubble flows. The solutions are presented for a general class of equations of state that includes the lattice QCD equations of state and may feature inhomogeneous temperature and corresponding density profiles.
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 describe fireballs that rehadronize from a perfectly fluid quark matter to a chemically frozen, multi-component hadron gas. In the hydrodynamics of these fireballs, we utilize the lattice QCD equation of state, however, we also apply non-relativistic kinematics for simplicity and clarity. Two new classes of exact, analytic solutions of fireball hydrodynamics are presented: the first class describes triaxially expanding, non-rotating ellipsoidal fireballs, while the second class of exact solutions corresponds to spheroidally symmetric, rotating fireballs. In both classes of solutions, we fin…
The Buda-Lund hydro model describes an expanding ellipsoidal fireball, and fits the observed elliptic flow and oscillating HBT radii successfully. Due to fluctuations in energy depositions, the fireball shape however fluctuates on an event-by-event basis. The transverse plane asymmetry can be translated into a series of multipole anisotropy coefficients. These anisotropies then result in measurable momentum-space anisotropies, to be measured with respect to their respective symmetry planes. In this paper we detail an extension of the Buda-Lund model to multipole anisotropies and investigate th…
The expansion of the fireball created in Au-Au collisions at 200 GeV is described in (3+1)-dimensional viscous hydrodynamics with shear and bulk viscosities. We present results for the transverse momentum spectra, the directed and elliptic flow and the interferometry radii.
We derive a system of moment-based dynamical equations that describe the 1+1d space-time evolution of a cylindrically symmetric massive gas undergoing boost-invariant longitudinal expansion. Extending previous work, we introduce an explicit degree of freedom associated with the bulk pressure of the system. The resulting form generalizes the ellipsoidal one-particle distribution function appropriate for massless particles to massive particles. Using this generalized form, we obtain a system of partial differential equations that can be solved numerically. In order to assess the performance of t…
A new method for evaluating spectra and correlations in the hydrodynamic approach is proposed. It is based on an analysis of Boltzmann equations (BE) in terms of probabilities for constituent particles to escape from the interacting system. The conditions of applicability of Cooper-Frye freeze-out prescription are considered within the method. The results are illustrated with a non-relativistic exact solution of BE for expanding spherical fireball as well as with approximate solutions for ellipsoidally expanding ones.
We solve analytically the ellipsoidally expanding fireball hydrodynamics with source terms in the momentum and the energy equations, using the non-relativistic approximation. We find that energy transport from high pt jets of gluons to the medium leads to a transient, exponential inflation of the fireballs created in high energy heavy ion collisions. In this transient, inflatory period, the slopes of the single particle spectra increase, while the HBT radius parameters decrease exponentially with time. This effect is shown to be similar to the development of the homogeneity of our Universe due…
We calculate the spectra of produced thermal photons in Au+Au collisions taking into account the nonequilibrium contribution to photon production due to finite shear viscosity. The evolution of the fireball is modeled by second-order as well as by divergence-type 2+1 dissipative hydrodynamics, both with an ideal equation of state and with one based on Lattice QCD that includes an analytical crossover. The spectrum calculated in the divergence-type theory is considerably enhanced with respect to the one calculated in the second-order theory, the difference being entirely due to differences in t…
A hybrid transport approach for the bulk evolution of viscous QCD matter produced in ultra-relativistic heavy-ion collisions is presented. The expansion of the dense deconfined phase of the reaction is modeled with viscous hydrodynamics while the dilute late hadron gas stage is described microscopically by the Boltzmann equation. The advantages of such a hybrid approach lie in the improved capability of handling large dissipative corrections in the late dilute phase of the reaction, including a realistic treatment of the non-equilibrium hadronic chemistry and kinetic freeze-out. By varying the…
We present a new class of exact fireball solutions of relativistic dissipative hydrodynamics. We describe new exact solutions both for the relativistic Navier-Stokes and for the Israel-Stewart theory, for arbitrary shear and bulk viscosities, as well as for other dissipative coefficients. The common property of these solutions is the presence of the relativistic Hubble flow. Our results generalize the recently found first solution in these classes, for an arbitrary temperature dependent speed of sound, shear and bulk viscosity, heat conduction and fluctuating initial temperature profiles. Thes…
The expansion of the fireball created in Au-Au collisions at 200GeV and Pb-Pb collisions at 2.76TeV is modelled using the relativistic viscous hydrodynamics. The experimentally observed interferometry radii are well reproduced. Additional pre-equilibrium flow improves slightly the results for the lower energies studied.
We present state-of-the-art calculations of viscous photon emission from nuclear collisions at RHIC and LHC. Fluctuating initial density profiles are evolved with event-by-event viscous hydrodynamics. Momentum spectra of thermal photons radiated by these explosively expanding fireballs and their $p_T$-differential anisotropic flow coefficients $v_n(p_T)$ are computed, both with and without accounting for viscous corrections to the standard thermal emission rates. Viscous corrections to the rates are found to have a larger effect on the $v_n$ coefficients than the viscous suppression of hydrody…
We perform viscous hydrodynamic calculations in 2+1 dimensions to investigate the influence of bulk viscosity on the viscous suppression of elliptic flow in non-central heavy-ion collisions at RHIC energies. Bulk and shear viscous effects on the evolution of radial and elliptic flow are studied with different model assumptions for the transport coefficients. We find that the temperature dependence of the relaxation time for the bulk viscous pressure, especially its critical slowing down near the quark-hadron phase transition at T_c, partially offsets effects from the strong growth of the bulk …
It is shown that the recently developed hybrid code VISHNU, which couples a relativistic viscous fluid dynamical description of the quark-gluon plasma (QGP) with a microscopic Boltzmann cascade for the late hadronic rescattering stage, yields an excellent description of charged and identified hadron spectra and elliptic flow measured in 200 A GeV Au+Au collisions at the Relativistic Heavy-Ion Collider (RHIC). Using initial conditions that incorporate event-by-event fluctuations in the initial shape and orientation of the collision fireball and values eta/s for the specific shear viscosity of t…
Heavy quark observables are applied to probe the initial energy density distribution with violation of longitudinal boost invariance produced in relativistic heavy-ion collisions. Using an improved Langevin model coupled to a (3+1)-dimensional viscous hydrodynamic model, we study the nuclear modification factor ($R_\text{AA}$), directed flow ($v_1$) and elliptic flow ($v_2$) coefficients of heavy mesons and their decayed electrons at the RHIC energy. We find that the counter-clockwise tilt of the nuclear matter in the reaction plane results in a positive (negative) heavy flavor $v_1$ in the ba…
The agreement of elliptic flow data at RHIC at central rapidity with the hydrodynamic model has led to the conclusion of very rapid thermalization. This conclusion is based on the intuitive argument that hydrodynamics, which assumes instantaneous local thermalization, produces the largest possible elliptic flow values and that the data seem to saturate this limit. We here investigate the question whether incompletely thermalized viscous systems may actually produce more elliptic flow than ideal hydrodynamics. Motivated by the extremely fast primordial longitudinal expansion of the reaction zon…
Study of thermal particle production is crucial to understand the space-time evolution of the fireball produced in high energy heavy-ion collisions. We consider thermal particle production within the framework of relativistic viscous hydrodynamics and employ recently obtained analytical solutions of higher-order viscous hydrodynamics with longitudinal Bjorken expansion to calculate the spectra of dileptons and photons. Using these analytical solutions, we constrain the allowed initial states by demanding positivity and reality of energy density throughout the evolution. Further, we compute the…
Following the Boz$\dot{\textrm{e}}$k-Wyskiel parametrization tilted initial condition, an alternative way to construct a longitudinal tilted fireball based on the Glauber collision geometry is presented. This longitudinal tilted initial condition combined with the Ideal-CLVisc (3 + 1)D hydrodynamic model, a nonvanishing directed flow coefficient $v_{1}$ in a wide rapidity range is observed. After comparing the model's results with experimentally observed data of directed flow coefficient $v_{1}(η)$ from $\sqrt{s_{NN}}$ = 200 GeV Cu + Cu, Au + Au collisions at RHIC energy to $\sqrt{s_{NN}}$ = 2…
The interferometric radii in the system formed in ultra-relativistic proton-lead collisions are investigated in a framework based on event-by-event 3+1 dimensional viscous hydrodynamics. We argue that the most central p-Pb collisions undergoing collective expansion behave similarly to the peripheral nucleus-nucleus collisions. The interferometric observables can serve as signatures of the formation of an extended fireball.
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+…
The fluid dynamics of a relativistic fireball with longitudinal and transverse expansion is described using a background-fluctuation splitting. Symmetry representations of azimuthal rotations and longitudinal boosts are used for a classification of initial state configurations and their fluid dynamic propagation in terms of a mode expansion. We develop an accurate and efficient numerical scheme based on the pseudo-spectral method to solve the resulting hyperbolic partial differential equations. Comparison to the analytically known Gubser solution underlines the high accuracy of this technique.…
We explore the effects of shear viscosity on the hydrodynamic evolution and final hadron spectra of Cu+Cu collisions at ultrarelativistic collision energies, using the newly developed (2+1)-dimensional viscous hydrodynamic code VISH2+1. Based on the causal Israel-Stewart formalism, this code describes the transverse evolution of longitudinally boost-invariant systems without azimuthal symmetry around the beam direction. Shear viscosity is shown to decelerate the longitudinal and accelerate the transverse hydrodynamic expansion. For fixed initial conditions, this leads to a longer quark-gluon p…
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 construct the QCD equation of state at finite chemical potentials including net baryon, electric charge, and strangeness based on the results of lattice QCD simulations and the hadron resonance gas model. The situation of strangeness neutrality and a fixed charge-to-baryon ratio, which resembles that of heavy nuclei, is considered for the application to relativistic heavy-ion collisions. This increases the values of baryon chemical potential compared to the case of vanishing strangeness and electric charge chemical potentials, modifying the fireball trajectory in the phase diagram. We perfo…
The specific shear viscosity (eta/s)_QGP of a Quark-Gluon-Plasma (QGP) at temperatures T_c < T < 2T_c is extracted from the centrality dependence of the eccentricity-scaled elliptic flow measured in ultra-relativistic heavy-ion collisions. Coupling viscous fluid dynamics for the QGP with a microscopic transport model for hadronic freeze-out we find that the eccentricity-scaled elliptic flow is a universal function of charged multiplicity per unit overlap area, (1/S)(dN_ch/dy), that depends only on the viscosity but not on the model used for computing the initial fireball eccentricity. Comparin…
Using a (2+1)-d viscous hydrodynamical model, we study the dependence of flow observables on the collision energy ranging from sqrt(s)=7.7 A GeV at the Relativistic Heavy Ion Collider (RHIC) to sqrt(s)=2760 A GeV at the Large Hadron Collider (LHC). With a realistic equation of state, Glauber model initial conditions and a small specific shear viscosity eta/s = 0.08, the differential charged hadron elliptic flow v_2^{ch}(p_T,sqrt(s)) is found to exhibit a very broad maximum as a function of sqrt(s) around top RHIC energy, rendering it almost independent of collision energy for 39 < sqrt(s) < 27…
With the new viscous hydrodynamic + hadron cascade hybrid code VISHNU, a rather precise (O(25%)) extraction of the QGP shear viscosity (eta/s)_QGP from heavy-ion elliptic flow data is possible if the initial eccentricity of the collision fireball is known with <5% accuracy. At this point, eccentricities from initial state models differ by up to 20%, leading to an O(100%) uncertainty for (eta/s)_QGP. It is shown that a simultaneous comparison of elliptic and triangular flow, v_2 and v_3, puts strong constraints on initial state models and can largely eliminate the present uncertainty in (eta/s)…
We determine the transverse system size of the initial non-equilibrium Glasma state and of the hydrodynamically evolving fireball as a function of produced charged particles in p+p, p+Pb and Pb+Pb collisions at the Large Hadron Collider. Our results are consistent with recent measurements of Hanbury-Brown-Twiss (HBT) radii by the ALICE collaboration. Azimuthal anisotropy coefficients v_n generated by combining the early time Glasma dynamics with viscous fluid dynamics in Pb+Pb collisions are in excellent agreement with experimental data for a wide range of centralities. In particular, event-by…
The stochastic dynamics of heavy quarks in the fireball produced in heavy-ion collisions is followed through numerical simulations based on the Langevin equation. The modification of the final p_T spectra (R_AA) of c and b quarks, hadrons and single-electrons with respect to pp collisions is studied. The transport coefficients are evaluated treating separately the contribution of soft and hard collisions. The initial heavy-quark spectra are generated according to NLO-pQCD, accounting for nuclear effects through recent nPDFs. The evolution of the medium is obtained from the output of two hydro-…
We study the nuclear modification of angular and momentum correlations between heavy quark pairs in ultrarelativistic heavy-ion collisions. The evolution of heavy quarks inside the thermalized medium is described via a modified Langevin approach that incorporates both elastic and inelastic interactions with the medium constituents. The spacetime evolution of the fireball is obtained from a (2+1)-dimensional viscous hydrodynamics simulation. The hadronization of heavy quarks is performed utilizing a hybrid model of fragmentation and coalescence. Our results show that the nuclear modification of…
Hydrodynamical description of the "Little Bang" in heavy ion collisions is surprisingly successful, mostly due to the very small viscosity of the Quark-Gluon plasma. In this paper we systematically study the propagation of small perturbations, also treated hydrodynamically. We start with a number of known techniques allowing for analytic calculation of the propagation of small perturbations on top of the expanding fireball. The simplest approximation is the "geometric acoustics", which substitutes the wave equation by mechanical equations for the propagating "phonons". Next we turn to the case…
In this paper we review recent progress in relativistic anisotropic hydrodynamics. We begin with a pedagogical introduction to the topic which takes into account the advances in our understanding of this topic since its inception. We consider both conformal and non-conformal systems and demonstrate how one can implement a realistic equation of state using a quasiparticle approach. We then consider the inclusion of non-spheroidal (non-ellipsoidal) corrections to leading-order anisotropic hydrodynamics and present the findings of the resulting second-order viscous anisotropic hydrodynamics frame…
We study the effects of initial state fluctuations on the dynamical evolution of heavy quarks inside a quark-gluon plasma created in relativistic heavy-ion collisions. The evolution of heavy quarks in QGP matter is described utilizing a modified Langevin equation that incorporates the contributions from both collisional and radiative energy loss. The spacetime evolution of the fireball medium is simulated with a (2+1)-dimensional viscous hydrodynamic model. We find that when the medium traversed by the heavy quark contains a fixed amount of energy, heavy quarks tend to lose more energy for gre…
We study the directed flow of identified particles in Au+Au collisions at $\sqrt{s_\text{NN}}=7.7$ to 62.4 GeV. The Glauber model is extended to include both a tilted deformation of the QGP fireball with respect to the longitudinal direction and a non-zero longitudinal flow velocity gradient in the initial state. By combining this improved initial condition with a (3+1)-dimensional viscous hydrodynamic model calculation, we obtain a satisfactory description of the transverse momentum spectra and the rapidity dependent directed flow coefficient of different hadron species. Our calculation indic…
We analyze the proton-lead collisions at the LHC energy of 5.02TeV in the three-stage approach, previously used to successfully describe the relativistic A-A collisions. The approach consists of the early phase, modeled with the Glauber model, the event-by-event viscous 3+1 dimensional (3+1 D) relativistic hydrodynamics, and the statistical hadronization at freeze-out. We show that features typical of collective dynamics, such as the harmonic flow and the ridge structures in the two-particle correlations in relative azimuth and pseudorapidity, may be naturally explained in our framework. In th…
In ultra-central heavy-ion collisions, anisotropic hydrodynamic flow is generated by density fluctuations in the initial state rather than by geometric overlap effects. For a given centrality class, the initial fluctuation spectrum is sensitive to the method chosen for binning the events into centrality classes. We show that sorting events by total initial entropy or by total final multiplicity yields event classes with equivalent statistical fluctuation properties, in spite of viscous entropy production during the fireball evolution. With this initial entropy-based centrality definition we ge…
