Bhatnagar-Gross-Krook (BGK) collision kernel is employed in the Boltzmann equation to formulate relativistic dissipative hydrodynamics. In this formulation, we find that there remains freedom of choosing a matching condition that affects the scalar transport in the system. We also propose a new collision kernel which, unlike BGK collision kernel, is valid in the limit of zero chemical potential and derive relativistic first-order dissipative hydrodynamics using it. We study the effects of this new formulation on the coefficient of bulk viscosity.
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 derive a first-order, stable and causal, relativistic hydrodynamic theory from the microscopic kinetic equation using the gradient expansion technique in a general frame. The general frame is introduced from the arbitrary matching conditions for hydrodynamic fields. The interaction is introduced in the relativistic Boltzmann equation through the momentum-dependent relaxation time approximation (MDRTA) with the proposed collision operator that preserves the conservation laws. We demonstrate here for the first time that not only the general frame choice, but also the momentum dependence of mi…
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 $η/s$.
In this article, we briefly review recent progress on hydrodynamic modeling and its implementations to relativistic heavy-ion collisions at RHIC and the LHC. The related topics include: 1) initial state fluctuations, final state correlations and event-by-event hydrodynamics, 2) extracting the QGP shear viscosity from flow data, 3) flow and hydrodynamics in 5.02 A TeV p+Pb collisions.
We present an exact solution of the relativistic Boltzmann equation for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse flow ("Gubser flow"). The resulting exact non-equilibrium dynamics is compared to 1st- and 2nd-order relativistic hydrodynamic approximations for various shear viscosity to entropy density ratios. This novel solution can be used to test the validity and accuracy of different hydrodynamic approximations in conditions similar to those generated in relativistic heavy-ion collisions.
We use the Crooks fluctuation theorem together with Zubarev hydrodynamics to develop a bottom-up theory of hydrodynamic fluctuations. We also use thermodynamic uncertainity relations to estimate bottom-up limits to dissipative transport coefficients.
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.
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 present a variational approach for relativistic ideal hydrodynamics interacting with electromagnetic fields. The momentum of fluid is introduced as the canonical conjugate variable of the position of a fluid element, which coincides with the conserved quantity derived from the Noether theorem. We further show that our formulation can reproduce the usual electromagnetic hydrodynamics which is obtained so as to satisfy the conservation of the inertia of fluid motion.
We generalize a previously known class of exact analytic solutions of relativistic perfect fluid hydrodynamics for the first time to arbitrary temperature-dependent Equation of State. We investigate special cases of this class of solutions, in particular, we present hydrodynamical solutions with the Equation of State determined from lattice QCD calculations. We discuss the phenomenological relevance of these solutions as well.
Three subjects are considered here: the relativistic hydrodynamics equations for a boost-invariant expanding fluid; the fuzzy bag model for the pressure which recently appeared in QCD phenomenology; and the early space-time evolution of the QCD matter, drawn from model studies, which can also be expected to arise in realistic fluid dynamics relevant to heavy ion collisions at LHC.
We present a first calculation of the dilepton yield and elliptic flow done with 3+1D viscous hydrodynamical simulations of relativistic heavy ion collisions at the top RHIC energy. A comparison with recent experimental data from the STAR collaboration is made.
We review recent progress in relativistic hydrodynamics, discussing causal and stable first-order hydrodynamics, known as BDNK theories, hydrodynamic attractors, as well as hydrodynamics near the chiral critical point and spin hydrodynamics.
We discuss some open problems in hydrodynamical approach to the relativistic heavy ion collisions. In particular, we propose a new, very simple alternative approach to the relativistic dissipative hydrodynamics of Israel and Stewart.
Hydrodynamics is a general theoretical framework for describing the long-time large-distance behaviors of various macroscopic physical systems, with its equations based on conservation laws such as energy-momentum conservation and charge conservation. Recently there has been significant interest in understanding the implications of angular momentum conservation for a corresponding hydrodynamic theory. In this work, we examine the key conceptual issues for such a theory in the relativistic regime where the orbital and spin components get entangled. We derive the equations for relativistic visco…
We study the evolution of the longitudinal expansion of an ideal fluid with finite electrical conductivity, which is subject to the EM fields. In the framework of resistive relativistic-magneto-hydrodynamic, we find an exact analytical solution for the EM fields and for the acceleration of the fluid.
We summarize the key ingredients of the recently proposed formalism of relativistic perfect-fluid hydrodynamics with spin. Based on the underlying kinetic theory definitions for the equilibrium distribution functions we obtain the evolution equations governing the system's expansion. Employing Bjorken symmetry we study the spin polarization dynamics of the system.
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.
Non-additive generalisation of relativistic anisotropic anisotropic hydrodynamics is described. In the particular case of 0+1 boost-invariant hydrodynamics additional entropy production due to non-additivity is calculated.
We present a new solution of relativistic hydrodynamics in 1+3 dimensions which depends on both the transverse coordinate and rapidity. At early times the flow expands dominantly longitudinally in a non-boost-invariant manner, and at late times it expands nearly spherically. These two regimes are shown to be related by symmetry. The effect of viscosity is also discussed.
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.
In this paper we analyze a 1+3 dimensional solution of relativistic hydrodynamics. We calculate momentum distribution and other observables from the solution and compare them to measurements from the Relativistic Heavy Ion Collider (RHIC). We find that the solution we analyze is compatible with the data. In the last several years many numerical models were tested, but it is the first time that an exact, parametric, 1+3 dimensional relativistic solution is compared to data.
We give a short review about the hydrodynamic model and its application to the elliptic flow phenomena and the pion interferometry in relativistic heavy ion collisions.
The quark gluon plasma is formed in heavy-ion collisions, and it can be described by solutions of relativistic hydrodynamics. In this paper we utilize perturbative hydrodynamics, where we study first order perturbations on top of a known solution. We investigate the perturbations on top of the Hubble flow. From this perturbative solution we can give the form of the particle emitting source and calculate observables of heavy-ion collisions. We describe the source function and the single-particle momentum spectra for a spherically symmetric solution.
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.
I summarize our recent work towards finding and utilizing analytic solutions of relativistic hydrodynamic. In the first part I discuss various exact solutions of the second-order conformal hydrodynamics. In the second part I compute flow harmonics $v_n$ analytically using the anisotropically deformed Gubser flow and discuss its dependence on $n$, $p_T$, viscosity, the chemical potential and the charge.
A new method for solving relativistic ideal hydrodynamics in (1+3)D is developed. Longitudinal and transverse radial flows are explicitly embedded and the hydrodynamic equations are reduced to a single equation for the transverse velocity field only, which is much more tractable. As an application we use the method to find analytically all possible solutions with power dependence on proper time and transverse radius. Possible application to the Relativistic Heavy Ion Collisions and possible generalizations of the method are discussed.
Shock wave solutions in anisotropic relativistic hydrodynamics are considered. A new phenomenon of anisotropy-related angular deflection of the incident flow by the shock wave front is described. Patterns of velocity and momentum transformation by the shock wave front are described.
We propose a new theory of second-order viscous relativistic hydrodynamics which does not impose any frame conditions on the choice of the hydrodynamic variables. It differs from Mueller-Israel-Stewart theory by including additional transient degrees of freedom, and its first-order truncation reduces to Bemfica-Disconzi-Noronha-Kovtun theory. Conditions for causality and stability are explicitly given in the conformal regime. As an illustrative example, we consider Bjorken flow solutions to our equations and identify variables which make a hydrodynamic attractor manifest.
Recent development of a hydrodynamic model is discussed by putting an emphasis on realistic treatment of the early and late stages in relativistic heavy ion collisions. The model, which incorporates a hydrodynamic description of the quark-gluon plasma with a kinetic approach of hadron cascades, is applied to analysis of elliptic flow data at the Relativistic Heavy Ion Collider energy. It is predicted that the elliptic flow parameter based on the hybrid model increases with the collision energy up to the Large Hadron Collider energy.
A new class of 3D anisotropic analytic solutions of relativistic hydrodynamics with constant pressure is found. We analyse, in particular, solutions corresponding to ellipsoidally symmetric expansion of finite systems into vacuum. They can be utilized for relativistic description of the system evolution in thermodynamic region near the softest point and at the final stage of the hydrodynamic expansion in A+A collisions. The solutions can be used also for testing of numerical hydrodynamic codes solving relativistic hydrodynamic equations for anisotropic expansion of finite systems
Generalizing the collision term in the relativistic Boltzmann equation to include nonlocal effects, and using Grad's 14-moment approximation for the single-particle distribution function, we derive evolution equations for the relativistic dissipative fluid dynamics and compare them with the corresponding equations obtained in the standard Israel-Stewart and related approaches. Significance of this generalization on hydrodynamic evolution is demonstrated in the framework of one-dimensional scaling expansion.
Relativistic hydrodynamics is essential to our current understanding of nucleus-nucleus collisions at ultrarelativistic energies (current experiments at the Relativistic Heavy Ion Collider, forthcoming experiments at the CERN Large Hadron Collider). This is an introduction to relativistic hydrodynamics for graduate students. It includes a detailed derivation of the equations, and a description of the hydrodynamical evolution of a heavy-ion collisions. Some knowledge of thermodynamics and special relativity is assumed.
Standard textbooks will state that hydrodynamics requires near-equilibrium to be applicable. Recently, however, out-of-equilibrium attractor solutions for hydrodynamics have been found in kinetic theory and holography in systems with a high degree of symmetry, suggesting the possibility of a genuine out-of-equilibrium formulation of hydrodynamics. This work demonstrates that attractor solutions also occur in non-conformal kinetic theory and spatially non-homogeneous systems, potentially having important implications for the interpretation of experimental data in heavy-ion and proton-proton col…
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.
Simulations of the viscous hydrodynamic model for relativistic heavy-ion collisions at RHIC and LHC energies are presented. Results for spectra, femtoscopy radii, and transverse momentum fluctuations are favorably compared to the experimental data. Effects of the local charge conservation on correlation observables are also studied.
The current status of the application of hydrodynamics to ultrarelativistic heavy ion collisions is reviewed. We elaborate on the arguments for strong transverse flow and rapid thermalization and discuss future applications and trends in hydrodynamics.
In a previous work, assuming that the nucleus can be treated as a perfect fluid, we have studied the propagation of perturbations in the baryon density. For a given equation of state we have derived a Korteweg - de Vries (KdV) equation from Euler and continuity equations in non-relativistic hydrodynamics. Here, using a more general equation of state, we extend our formalism to relativistic hydrodynamics.
Freeze-out of particles in relativistic hydrodynamics is considered across a 3-dimensional space-time hypersurface. The conservation laws for time-like parts of the freeze-out hypersurface require different values of temperature, baryonic chemical potential and flow velocity in the fluid and in the final particle spectra. We analyze this freeze-out discontinuity and its connection to the shock-wave phenomena in relativistic hydrodynamics.
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.
Relativistic hydrodynamics for ideal and viscous fluids is discussed as a tool to describe relativistic heavy-ion collisions and to extract transport properties of the quark-gluon plasma from experimentally measured hadron momentum spectra.
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.
Formalism to calculate the hydrodynamic fluctuations by applying the Onsager theory to the relativistic Navier-Stokes equation is already known. In this work, we calculate hydrodynamic-fluctuations within the framework of the second order hydrodynamics of Müller, Israel and Stewart and its generalization to the third order. We have also calculated the fluctuations for several other causal hydrodynamical equations. We show that the form for the Onsager-coefficients and form of the correlation-functions remains same as those obtained by the relativistic Navier-Stokes equation and it does not dep…
In this proceeding, we will briefly review our recent progress on implementing deep learning to relativistic hydrodynamics. We will demonstrate that a successfully designed and trained deep neural network, called {\tt stacked U-net}, can capture the main features of the non-linear evolution of hydrodynamics, which could also rapidly predict the final profiles for various testing initial conditions.
Relativistic hydrodynamics is a powerful tool to simulate the evolution of the quark gluon plasma (QGP) in relativistic heavy ion collisions. Using 10000 initial and final profiles generated from 2+1-d relativistic hydrodynamics VISH2+1 with MC-Glauber initial conditions, we train a deep neural network based on stacked U-net, and use it to predict the final profiles associated with various initial conditions, including MC-Glauber, MC-KLN and AMPT and TRENTo. A comparison with the VISH2+1 results shows that the network predictions can nicely capture the magnitude and inhomogeneous structures of…
We obtain general analytical solutions of third-order viscous hydrodynamic equations for Bjorken and Gubser flows in systems with vanishing bulk viscosity and chemical potential, and having a constant shear relaxation time. We also analytically characterize and determine the hydrodynamic attractors for such systems by studying the universal behavior of these solutions at both early and late times. Finally we study the properties of these hydrodynamic attractors for transport coefficients obtained from relativistic kinetic theory in the relaxation-time approximation.
General features of the formalism describing hydrodynamic evolution of transversally thermalized matter possibly produced at the very early stages of ultra-relativistic heavy-ion collisions are presented. Thermodynamical consistency of the model is emphasized. The covariant formulas for the moments of the phase-space distribution function are derived. The simple model for the transition from purely transverse to standard perfect-fluid hydrodynamics is proposed.
We study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. In a previous work we have derived a KdV equation from Euler and continuity equations in non-relativistic hydrodynamics. In the present contribution we extend our formalism to relativistic fluids. We present results for a given equation of state, which is based on quantum hadrodynamics (QHD).
Hydrodynamic behavior is a general feature of interacting systems with many degrees of freedom constrained by conservation laws. To date hydrodynamic scaling in relativistic quantum systems has been observed in many high energy settings, from cosmic ray detections to accelerators, with large particle multiplicity final states. Here we show first evidence for the emergence of hydrodynamic scaling in the dynamics of a relativistic quantum field theory. We consider a simple scalar $λϕ^4$ model in 1+1 dimensions in the Hartree approximation and study the dynamics of two colliding kinks at relativi…
The matter created in relativistic heavy ion collisions is fairly well described by ideal hydrodynamics, and somewhat better described by viscous hydrodynamics. To this point, most viscous calculations have been two-dimensional, based on an assumption of Bjorken boost invariance along the beam axis. Here, first results are presented for a fully three-dimensional viscous model. The model is described and tests of the numerical accuracy of the code are presented. Two- and three-dimensional runs are compared, and modest changes are observed for mid-rapidity observables at the highest RHIC (Relati…
Discontinuities in relativistic hydrodynamics - shock waves and freeze-out shocks - are considered across both space-like and time-like hypersurfaces. We analyze the peculiar features of the freeze-out discontinuities and their connection to the shock-wave phenomena.
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.
