The success of relativistic hydrodynamics as an essential part of the phenomenological description of heavy-ion collisions at RHIC and the LHC has motivated a significant body of theoretical work concerning its fundamental aspects. Our review presents these developments from the perspective of the underlying microscopic physics, using the language of quantum field theory, relativistic kinetic theory, and holography. We discuss the gradient expansion, the phenomenon of hydrodynamization, as well as several models of hydrodynamic evolution equations, highlighting the interplay between collective long-lived and transient modes in relativistic matter. Our aim to provide a unified presentation of this vast subject-which is naturally expressed in diverse mathematical languages-has also led us to include several new results on the large-order behaviour of the hydrodynamic gradient expansion.

The assumption of simultaneous chemical and thermal freeze-outs of the hadron gas leads to a surprisingly accurate, albeit entirely conventional, explanation of the recently-measured RHIC p ⊥ -spectra. The original thermal spectra are supplied with secondaries from cascade decays of all resonances, and subsequently folded with a suitably parameterized expansion involving longitudinal and transverse flow. The predictions of this thermal approach, with various parametrizations for the expansion, are in a striking quantitative agreement with the data in the whole available range of 0 ≤ p ⊥ ≤ 3.5GeV. 25.75.Dw, 25.75.Ld In this Letter we offer a very simple explanation of the p ⊥ -spectra recently measured at RHIC [1][2][3]. Our approach has the following ingredients: i) simultaneous chemical and thermal freeze-outs, with the hadron distributions given by the thermal model; in other words, hadrons decouple completely when the thermodynamic parameters reach the freezing conditions, and no particle rescattering after freeze-out is present, ii) these thermal distributions are folded with a suitably parameterized hydrodynamic expansion, involving longitudinal and transverse flow, finally, iii) feeding from resonances, including cascades, is incorporated in a complete way.So far, the thermal approach has been applied successfully in studies of particle ratios measured in relativistic heavy-ion collisions at AGS and SPS [4][5][6][7][8][9][10]. Quite recently, it has also been shown that particle ratios measured at RHIC may be equally well described in the framework of such models [11][12][13]. Description of hadronic p ⊥ -spectra in thermal models is more involved, since the spectra are affected by decays of resonances, hydrodynamic flow, and possibly by other phenomena occurring during the alleged phase transition from the quark-gluon plasma to a hadron gas [14]. The model results presented in this Letter are in a surprising agreement with the experiment in the entire range of the data, 0 ≤ p ⊥ ≤ 3.5GeV, as can be seen in Fig. 1. The model has two free parameters: one controlling the size of the system (overall normalization of the spectra), and the other one the transverse flow. We test two different models (parametrizations) for the freeze-out hypersurface and the hydrodynamic expansion. Both combine the Bjorken expansion [15] with transverse flow [16,17], and follow the spirit of Refs. [18][19][20][21][22].The first model (model I) assumes that the freezeout takes place at a fixed value of the invariant time, τ = t 2 − r 2 z − r 2 x − r 2 y = const, which means that, due to time dilation, the particles in the fluid elements moving farther away from the collision center decouple later than the particles in the fluid elements remaining at rest in the center-of-mass system of the colliding nuclei. Furthermore, we assume that the four-velocity of expansion is proportional to the coordinate,The freeze-out hypersurface is parameterized as [20]where α is the rapidity of the fluid element (v z = r z /t = tanh α ), whereas ...

PrefaceThe experimental studies of strongly interacting matter produced in ultrarelativistic heavy-ion collisions belong to the avant-garde of contemporary highenergy physics. This new and vastly developing field requires theoretical understanding how new experimental phenomena are related to the physical properties of the created system. This book delivers foundations of such understanding -it shows the links between basic theoretical concepts, discussed gradually from the elementary to more advanced level, and the results of experiments. In this way, I hope, experimentalists may learn more about the foundations of the models used by them to fit and interpret the data, while theoreticians may learn more about practical applications of their ideas.The book emphasizes the role played in the interpretation of the experimental results by thermodynamics, relativistic hydrodynamics, and relativistic kinetic theory. These frameworks are used to analyze the soft hadron production, i.e., the production of hadrons with relatively small transverse momenta with respect to the collision axis. Soft hadrons contribute to more than 90% of the produced particles and their measured properties reveal information about the bulk properties of the very hot and dense system formed at the early stages of collisions -an interacting quark-gluon plasma.A discussion of hard phenomena, i.e., the production of hadrons with large transverse momenta, has been omitted (except for the general comments concerning the jet quenching). Certainly, the role of hard processes is becoming more and more important with the increasing beam energy, however, their discussion would double the size of the present book.The successful application of the perfect-fluid hydrodynamics in description of ultra-relativistic heavy-ion collisions allows us to establish a uniform picture of these complicated processes. In some sense we are lucky that such complex systems may be described within a concise and well-defined framework. The perfect-fluid hydrodynamics, combined with the modeling of the initial state by the Glauber model or the color glass condensate on one side, and supplemented by the kinetic simulations of the freeze-out process on the other side, forms the foundation of an approach that may be regarded as the standard model of ultra-relativistic heavy-ion collisions. Yet, several shortcomings of the hydrodynamic approach indicate directions for many current and new investigations. To list a few: The problem of very early equilibration of matter formed in heavy-ion collisions is discussed in a very broad context including more elementary processes such as e + e − annihilation. Difficulties connected with the correct description of the correlations form a challenge for consistent modeling of the momentum and spacetime distributions of particles. Finally, the role of the viscosity and other dissipative effects should be elucidated. By the way, the recent developments in the field of dissipative hydrodynamics are examples of the permanent progress that is taki...

Using the conservation laws for charge, energy, momentum, and angular momentum, we derive hydrodynamic equations for the charge density, local temperature, and fluid velocity, as well as for the polarization tensor, starting from local equilibrium distribution functions for particles and antiparticles with spin 1 /2. The resulting set of differential equations extend the standard picture of perfect-fluid hydrodynamics with a conserved entropy current in a minimal way. This framework can be used in space-time analyses of the evolution of spin and polarization in various physical systems including high-energy nuclear collisions. We demonstrate that a stationary vortex, which exhibits vorticity-spin alignment, corresponds to a special solution of the spin-hydrodynamical equations.

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 expectations about the behavior of matter produced at the early stages of relativistic heavy-ion collisions. The equilibration is identified with the processes of entropy production.

We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times, anisotropic hydrodynamics provides a very good approximation to the exact relaxation time approximation solution.

We exactly solve the relaxation-time approximation Boltzmann equation for a system which is transversely homogeneous and undergoing boost-invariant longitudinal expansion. We compare the resulting exact numerical solution with approximate solutions available in the anisotropic hydrodynamics and second order viscous hydrodynamics frameworks. In all cases studied, we find that the anisotropic hydrodynamics framework is a better approximation to the exact solution than traditional viscous hydrodynamical approaches.The application of relativistic viscous hydrodynamics is important in a wide variety of situations including, for example, the dynamics of high energy astrophysical plasmas and the quark-gluon plasma created in relativistic heavy-ion collisions. Since the seminal work of Israel and Stewart [1,2] there have been many papers that have addressed the questions of how to apply and systematically improve relativistic viscous hydrodynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Recently, a new framework called anisotropic hydrodynamics (aHydro) has emerged for describing the nonequilibrium dynamics of relativistic systems [22][23][24][25][26][27][28][29].

We investigate the hydrodynamic evolution of the system formed in ultrarelativistic heavy-ion collisions and find that an appropriate choice of the initial condition, specifically a simple two-dimensional Gaussian profile for the transverse energy, in conjunction with a realistic equation of state, leads to a uniform description of soft observables measured at the relativistic heavy-ion collider. In particular, the transverse-momentum spectra, the elliptic-flow, and the Hanbury-Brown-Twiss correlation radii, including the ratio Rout/Rside as well as the dependence of the radii on the azimuthal angle, are all properly described.