Abstract

Relativistic hydrodynamics has been quite successful in explaining the collective behaviour of the QCD matter produced in high energy heavy-ion collisions at RHIC and LHC. We briefly review the latest developments in the hydrodynamical modeling of relativistic heavy-ion collisions. Essential ingredients of the model such as the hydrodynamic evolution equations, dissipation, initial conditions, equation of state, and freeze-out process are reviewed. We discuss observable quantities such as particle spectra and anisotropic flow and effect of viscosity on these observables. Recent developments such as event-by-event fluctuations, flow in small systems (proton-proton and proton-nucleus collisions), flow in ultracentral collisions, longitudinal fluctuations, and correlations and flow in intense magnetic field are also discussed.

1. Introduction

The existence of both confinement and asymptotic freedom in Quantum Chromodynamics (QCD) has led to many speculations about its thermodynamic and transport properties. Due to confinement, the nuclear matter must be made of hadrons at low energies; hence it is expected to behave as a weakly interacting gas of hadrons. On the other hand, at very high energies, asymptotic freedom implies that quarks and gluons interact only weakly and the nuclear matter is expected to behave as a weakly coupled gas of quarks and gluons. In between these two configurations there must be a phase transition, where the hadronic degrees of freedom disappear, and a new state of matter, in which the quark and gluon degrees of freedom manifest directly over a certain volume, is formed. This new phase of matter, referred to as quark-gluon plasma (QGP), is expected to be created when sufficiently high temperatures or densities are reached [1–3].

The QGP is believed to have existed in the very early universe (a few microseconds after the Big Bang) or some variant of which possibly still exists in the inner core of a neutron star, where it is estimated that the density can reach values ten times higher than those of ordinary nuclei. It was conjectured theoretically that such extreme conditions can also be realised on earth, in a controlled experimental environment, by colliding two heavy nuclei with ultrarelativistic energies [4]. This may transform a fraction of the kinetic energies of the two colliding nuclei into heating the QCD vacuum within an extremely small volume, where temperatures million times hotter than the core of the sun may be achieved.

With the advent of modern accelerator facilities, ultrarelativistic heavy-ion collisions have provided an opportunity to systematically create and study different phases of the bulk nuclear matter. It is widely believed that the QGP phase is formed in heavy-ion collision experiments at Relativistic Heavy-Ion Collider (RHIC) located at Brookhaven National Laboratory, USA, and Large Hadron Collider (LHC) at European Organization for Nuclear Research (CERN), Geneva. A number of indirect evidences found at the Super Proton Synchrotron (SPS) at CERN strongly suggested the formation of a “new state of matter” [5], but quantitative and clear results were only obtained at RHIC energies [6–14] and recently at LHC energies [15–18]. The regime with relatively large baryon chemical potential will be probed by the upcoming experimental facilities like Facility for Antiproton and Ion Research (FAIR) at GSI, Darmstadt. An illustration of the QCD phase diagram and the regions probed by these experimental facilities is shown in Figure 1 [19].

Figure 1 

Schematic phase diagram of the QCD matter. The net baryon density on -axis is normalized to that of the normal nuclear matter [19].

It is possible to create hot and dense nuclear matter with very high energy densities in relatively large volumes by colliding ultrarelativistic heavy ions. In these conditions, the nuclear matter created may be close to (local) thermodynamic equilibrium, providing the opportunity to investigate the various phases and the thermodynamic and transport properties of QCD. It is important to note that even though it appears that a deconfined state of matter is formed in these colliders, investigating and extracting the transport properties of QGP from heavy-ion collisions is not an easy task, since it cannot be observed directly. Experimentally, it is only feasible to measure energy and momenta of the particles produced in the final stages of the collision, when nuclear matter is already relatively cold and noninteracting. Hence, in order to study the thermodynamic and transport properties of the QGP, the whole heavy-ion collision process from the very beginning till the end has to be modeled: starting from the stage where two highly Lorentz contracted heavy nuclei collide with each other, the formation and thermalization of the QGP or deconfined phase in the initial stages of the collision, its subsequent space-time evolution, the phase transition to the hadronic or confined phase of matter, and, eventually, the dynamics of the cold hadronic matter formed in the final stages of the collision. The different stages of ultrarelativistic heavy-ion collisions are schematically illustrated in Figure 2 [20].

Figure 2 

Various stages of ultrarelativistic heavy-ion collisions [20].

Assuming that thermalization is achieved in the early stages of heavy-ion collisions and that the interaction between the quarks is strong enough to maintain local thermodynamic equilibrium during the subsequent expansion, the time evolution of the QGP and hadronic matter can be described by the laws of fluid dynamics [21–24]. Fluid dynamics, also loosely referred to as hydrodynamics, is an effective approach through which a system can be described by macroscopic variables, such as local energy density, pressure, temperature, and flow velocity. The most appealing feature of fluid dynamics is the fact that it is simple and general. It is simple in the sense that all the information of the system is contained in its thermodynamic and transport properties, that is, its equation of state and transport coefficients. Fluid dynamics is also general because it relies on only one assumption: the system remains close to local thermodynamic equilibrium throughout its evolution. Although the hypothesis of proximity to local equilibrium is quite strong, it saves us from making any further assumption regarding the description of the particles and fields, their interactions, the classical or quantum nature of the phenomena involved, and so forth. Hydrodynamic analysis of the spectra and azimuthal anisotropy of particles produced in heavy-ion collisions at RHIC [25, 26] and recently at LHC [27, 28] suggests that the matter formed in these collisions is strongly coupled quark-gluon plasma (sQGP).

Application of viscous hydrodynamics to high energy heavy-ion collisions has evoked widespread interest ever since a surprisingly small value of was estimated from the analysis of the elliptic flow data [25]. It is interesting to note that, in the strong coupling limit of a large class of holographic theories, the value of the shear viscosity to entropy density ratio . Kovtun, Son, and Starinets (KSS) conjectured this strong coupling result to be the absolute lower bound for all fluids; that is, [29, 30]. This specific combination of hydrodynamic quantities, , accounts for the difference between momentum and charge diffusion such that even though the diffusion constant goes to zero in the strong coupling limit, the ratio remains finite [31]. Similar result for a lower bound also follows from the quantum mechanical uncertainty principle. The kinetic theory prediction for viscosity, , suggests that low viscosity corresponds to short mean free path. On the other hand, the uncertainty relation implies that the product of the mean free path and the average momentum cannot be arbitrarily small; that is, . For a weakly interacting relativistic Bose gas, the entropy per particle is given by . This leads to which is very close to the lower KSS bound.

Indeed, the estimated of QGP was so close to the KSS bound that it led to the claim that the matter formed at RHIC was the most perfect fluid ever observed. A precise estimate of is vital to the understanding of the properties of the QCD matter and is presently a topic of intense investigation; see [32] and references therein for more details. In this review, we shall discuss the general aspects of the formulation of relativistic fluid dynamics and its application to the physics of high energy heavy-ion collisions. Along with these general concepts, we shall also discuss here some of the recent developments in the field. Among the recent developments, the most striking feature is the experimental observation of flow like pattern in the particle azimuthal distribution of high multiplicity proton-proton (p-p) and proton-lead (p-Pb) collisions. We will discuss in this review the success of hydrodynamics model in describing these recent experimental measurements by assuming hydrodynamics flow of small systems.

The review is organized as follows. In Section 2 we discuss the general formalism of a causal theory of relativistic dissipative fluid dynamics. Section 3 deals with the initial conditions necessary for the modeling of relativistic heavy-ion, p-p, and p-Pb collisions. In Section 4, we discuss some models of preequilibrium dynamics employed before hydrodynamic evolution. Section 5 briefly covers various equations of state, necessary to close the hydrodynamic equations. In Section 6, particle production mechanism via Cooper-Frye freeze-out and anisotropic flow generation is discussed. In Section 7, we discuss models of hadronic rescattering after freeze-out and contribution to particle spectra and flow from resonance decays. Section 8 deals with the extraction of transport coefficients from hydrodynamic analysis of flow data. Finally, in Section 9, we discuss recent developments in the hydrodynamic modeling of relativistic collisions.

In this review, unless stated otherwise, all physical quantities are expressed in terms of natural units, where , with , where is the Planck constant, is the velocity of light in vacuum, and is the Boltzmann constant. Unless stated otherwise, the space-time is always taken to be Minkowskian, where the metric tensor is given by . Apart from Minkowskian coordinates , we will also regularly employ Milne coordinate system or , with proper time , the radial coordinate , the azimuthal angle , and space-time rapidity . Hence, , , , and . For the coordinate system , the metric becomes , whereas for , the metric is . Roman letters are used to indicate indices that vary from 1 to 3 and Greek letters are used for indices that vary from 0 to 3. Covariant and contravariant four-vectors are denoted as and , respectively. The notation represents scalar product of a covariant and a contravariant four-vector. Tensors without indices shall always correspond to Lorentz scalars. We follow Einstein summation convention, which states that repeated indices in a single term are implicitly summed over all the values of that index.

We denote the fluid four-velocity by and the Lorentz contraction factor by . The projector onto the space orthogonal to is defined as . Hence, satisfies the conditions with trace . The partial derivative can then be decomposed asIn the fluid rest frame, reduces to the time derivative and reduces to the spacial gradient. Hence, the notation is also commonly used. We also frequently use the symmetric, antisymmetric, and angular brackets notations defined as where is the traceless symmetric projection operator orthogonal to satisfying the conditions .

2. Relativistic Fluid Dynamics

The physical characterization of a system consisting of many degrees of freedom is in general a nontrivial task. For instance, the mathematical formulation of a theory describing the microscopic dynamics of a system containing a large number of interacting particles is one of the most challenging problems of theoretical physics. However, it is possible to provide an effective macroscopic description, over large distance and time scales, by taking into account only the degrees of freedom which are relevant at these scales. This is a consequence of the fact that on macroscopic distance and time scales the actual degrees of freedom of the microscopic theory are imperceptible. Most of the microscopic variables fluctuate rapidly in space and time; hence only average quantities resulting from the interactions at the microscopic level can be observed on macroscopic scales. These rapid fluctuations lead to very small changes of the average values and hence are not expected to contribute to the macroscopic dynamics. On the other hand, variables that do vary slowly, such as the conserved quantities, are expected to play an important role in the effective description of the system. Fluid dynamics is one of the most common examples of such a situation. It is an effective theory describing the long wavelength, low frequency limit of the underlying microscopic dynamics of a system.

A fluid is defined as a continuous system in which every infinitesimal volume element is assumed to be close to thermodynamic equilibrium and to remain near equilibrium throughout its evolution. Hence, in other words, in the neighbourhood of each point in space, an infinitesimal volume called fluid element is defined in which the matter is assumed to be homogeneous; that is, any spatial gradients can be ignored, and it is described by a finite number of thermodynamic variables. This implies that each fluid element must be large enough, compared to the microscopic distance scales, to guarantee the proximity to thermodynamic equilibrium, and, at the same time, must be small enough, relative to the macroscopic distance scales, to ensure the continuum limit. The coexistence of both continuous (zero volume) and thermodynamic (infinite volume) limits within a fluid volume might seem paradoxical at first glance. However, if the microscopic and the macroscopic length scales of the system are sufficiently far apart, it is always possible to establish the existence of a volume that is small enough compared to the macroscopic scales and at the same time large enough compared to the microscopic ones. Here, we will assume the existence of clear separation between microscopic and macroscopic scales to guarantee the proximity to local thermodynamic equilibrium.

Relativistic fluid dynamics has been quite successful in explaining the various collective phenomena observed in astrophysics, cosmology, and the physics of high energy heavy-ion collisions. In cosmology and certain areas of astrophysics, one needs a fluid dynamics formulation consistent with the General Theory of Relativity. On the other hand, a formulation based on the Special Theory of Relativity is quite adequate to treat the evolution of the strongly interacting matter formed in high energy heavy-ion collisions when it is close to a local thermodynamic equilibrium. In fluid dynamical approach, although no detailed knowledge of the microscopic dynamics is needed, knowledge of the equation of state relating pressure, energy density, and baryon density is required. The collective behaviour of the hot and dense quark-gluon plasma created in ultrarelativistic heavy-ion collisions has been studied quite extensively within the framework of relativistic fluid dynamics. In application of fluid dynamics, it is natural to first employ the simplest version which is ideal hydrodynamics which neglects the viscous effects and assumes that local equilibrium is always perfectly maintained during the fireball expansion. Microscopically, this requires that the microscopic scattering time be much shorter than the macroscopic expansion (evolution) time. In other words, ideal hydrodynamics assumes that the mean free path of the constituent particles is much smaller than the system size. However, as all fluids are dissipative in nature due to the quantum mechanical uncertainty principle [33], the ideal fluid results serve only as a benchmark when dissipative effects become important.

When discussing the application of relativistic dissipative fluid dynamics to heavy-ion collision, one is faced with yet another predicament: the theory of relativistic dissipative fluid dynamics is not yet conclusively established. In fact, introducing dissipation in relativistic fluids is not at all a trivial task and still remains one of the important topics of research in high energy physics. Ideal hydrodynamics assumes that local thermodynamic equilibrium is perfectly maintained and each fluid element is homogeneous; that is, spatial gradients are absent (zeroth order in gradient expansion). If this is not satisfied, dissipative effects come into play. The earliest theoretical formulations of relativistic dissipative hydrodynamics, also known as first-order theories, are due to Eckart [34] and Landau and Lifshitz [35]. However, these formulations, collectively called relativistic Navier-Stokes (NS) theory, suffer from acausality and numerical instability. The reason for the acausality is that in the gradient expansion the dissipative currents are linearly proportional to gradients of temperature, chemical potential, and velocity, resulting in parabolic equations. Thus, in Navier-Stokes theory, the gradients have an instantaneous influence on the dissipative currents. Such instantaneous effects tend to violate causality and cannot be allowed in a covariant setup, leading to the instabilities investigated in [36–38].

The second-order Israel-Stewart (IS) theory [39] restores causality but may not guarantee stability [40]. The acausality problems were solved by introducing a time delay in the creation of the dissipative currents from gradients of the fluid dynamical variables. In this case, the dissipative quantities become independent dynamical variables obeying equations of motion which describe their relaxation towards their respective Navier-Stokes values. The resulting equations are hyperbolic in nature which preserves causality. Israel-Stewart theory has been widely applied to ultrarelativistic heavy-ion collisions in order to describe the time evolution of the QGP and the subsequent freeze-out process of the hadron resonance gas. Although IS hydrodynamics has been quite successful in modeling relativistic heavy-ion collisions, there are several inconsistencies and approximations in its formulation which prevent proper understanding of the thermodynamic and transport properties of the QGP. Moreover, the second-order IS theory can be derived in several ways, each leading to a different set of transport coefficients. Therefore, in order to quantify the transport properties of the QGP from experiment and confirm the claim that it is indeed the most perfect fluid ever created, the theoretical foundations of relativistic dissipative fluid dynamics must be first addressed and clearly understood. In this section, we review the basic aspects of thermodynamics and discuss the formulation of relativistic fluid dynamics from a phenomenological perspective. The salient features of kinetic theory in the context of fluid dynamics will also be discussed.

2.1. Thermodynamics

Thermodynamics is an empirical description of the macroscopic or large-scale properties of matter and it makes no hypotheses about the small-scale or microscopic structure. It is concerned only with the average behaviour of a very large number of microscopic constituents, and its laws can be derived from statistical mechanics. A thermodynamic system can be described in terms of a small set of extensive variables, such as volume , the total energy , entropy , and number of particles , of the system. Thermodynamics is based on four phenomenological laws that explain how these quantities are related and how they change with time [41–43]:(i)Zeroth Law: if two systems are both in thermal equilibrium with a third system then they are in thermal equilibrium with each other. This law helps define the notion of temperature(ii)First Law: all the energy transfers must be accounted for to ensure the conservation of the total energy of a thermodynamic system and its surroundings. This law is the principle of conservation of energy(iii)Second Law: an isolated physical system spontaneously evolves towards its own internal state of thermodynamic equilibrium. Employing the notion of entropy, this law states that the change in entropy of a closed thermodynamic system is always positive or zero(iv)Third Law: it is also known as Nernst’s heat theorem and states that the difference in entropy between systems connected by a reversible process is zero in the limit of vanishing temperature. In other words, it is impossible to reduce the temperature of a system to absolute zero in a finite number of operations

The first law of thermodynamics postulates that the changes in the total energy of a thermodynamic system must result from (1) heat exchange, (2) the mechanical work done by an external force, and (3) particle exchange with an external medium. Hence, the conservation law relating the small changes in state variables, , , and iswhere and are the pressure and chemical potential, respectively, and is the amount of heat exchange.

The heat exchange takes into account the energy variations due to changes of internal degrees of freedom which are not described by the state variables. The heat itself is not a state variable, since it can depend on the past evolution of the system and may take several values for the same thermodynamic state. However, when dealing with reversible processes (in time), it becomes possible to assign a state variable related to heat. This variable is the entropy, , and is defined in terms of the heat exchange as , with the temperature being the proportionality constant. Then, when considering variations between equilibrium states that are infinitesimally close to each other, it is possible to write the first law of thermodynamics in terms of differentials of the state variables:Hence, using (5), the intensive quantities, , and , can be obtained in terms of partial derivatives of the entropy as

The entropy is mathematically defined as an extensive and additive function of the state variables, which means thatDifferentiating both sides with respect to , we obtainwhich holds for any arbitrary value of . Setting and using (6), we obtain the so-called Euler’s relation:Using Euler’s relation, (9), along with the first law of thermodynamics, (5), we arrive at the Gibbs-Duhem relation:

In terms of energy, entropy, and number of densities defined as , , and , respectively, Euler’s relation, (9), and Gibbs-Duhem relation, (10), reduce toDifferentiating (11) and using (12), we obtain the relation analogous to first law of thermodynamics:It is important to note that all the densities defined above are intensive quantities.

The equilibrium state of a system is defined as a stationary state, where the extensive and intensive variables of the system do not change. We know from the second law of thermodynamics that the entropy of an isolated thermodynamic system must either increase or remain constant. Hence, if a thermodynamic system is in equilibrium, the entropy of the system, being an extensive variable, must remain constant. On the other hand, for a system that is out of equilibrium, the entropy must always increase. This is an extremely powerful concept that will be extensively used in this section to constrain and derive the equations of motion of a dissipative fluid. This concludes a brief outline of the basics of thermodynamics; for a more detailed review, see [43]. In the next section, we introduce and derive the equations of relativistic ideal fluid dynamics.

2.2. Relativistic Ideal Fluid Dynamics

An ideal fluid is defined by the assumption of local thermal equilibrium; that is, all fluid elements must be exactly in thermodynamic equilibrium [35, 44]. This means that, at each space-time coordinate of the fluid , there can be assigned a temperature , a chemical potential , and a collective four-velocity field:The proper time increment is given by the line elementwhere . This implies thatwhere . In the nonrelativistic limit, we obtain . It is important to note that the four-vector only contains three independent components, since it obeys the relationThe quantities , , and are often referred to as the primary fluid dynamical variables.

The state of a fluid can be completely specified by the densities and currents associated with conserved quantities, that is, energy, momentum, and (net) particle number. For a relativistic fluid, the state variables are the energy-momentum tensor, , and the (net) particle four-current, . To obtain the general form of these currents for an ideal fluid, we first define the local rest frame (LRF) of the fluid. In this frame, , and the energy-momentum tensor, , the (net) particle four-current, , and the entropy four-current, , should have the characteristic form of a system in static equilibrium. In other words, in local rest frame, there is no flow of energy , the force per unit surface element is isotropic , and there is no particle and entropy flow ( and ). Consequently, the energy-momentum tensor, particle four-current, and entropy four-current in this frame take the following simple forms:

For an ideal relativistic fluid, the general form of the energy-momentum tensor, , (net) particle four-current, , and the entropy four-current, , has to be built out of the hydrodynamic tensor degrees of freedom, namely, the vector, , and the metric tensor, . Since should be symmetric and transform as a tensor and and should transform as a vector, under Lorentz transformations, the most general form allowed is thereforeIn the local rest frame, . Hence, in this frame, (19) takes the formBy comparing the above equation with the corresponding general expressions in the local rest frame, (18), one obtains the following expressions for the coefficients:The conserved currents of an ideal fluid can then be expressed aswhere is the projection operator on the three-space orthogonal to and satisfies the following properties of an orthogonal projector:

The dynamical description of an ideal fluid is obtained using the conservation laws of energy, momentum, and (net) particle number. These conservation laws can be mathematically expressed using the four-divergences of energy-momentum tensor and particle four-current which leads to the following equations:where the partial derivative transforms as a covariant vector under Lorentz transformations. Using the four-velocity, , and the projection operator, , the derivative, , can be projected along and orthogonal to :Projection of energy-momentum conservation equation along and orthogonal to together with the conservation law for particle number leads to the equations of motion of ideal fluid dynamics:where . It is important to note that an ideal fluid is described by four fields, , , , and , corresponding to six independent degrees of freedom. The conservation laws, on the other hand, provide only five equations of motion. The equation of state of the fluid, , relating the pressure to other thermodynamic variables has to be specified to close this system of equations. The existence of equation of state is guaranteed by the assumption of local thermal equilibrium and hence the equations of ideal fluid dynamics are always closed.

2.3. Covariant Thermodynamics

In the following, we rewrite the equilibrium thermodynamic relations derived in Section 2.1, (11), (12), and (13), in a covariant form [39, 45]. For this purpose, it is convenient to introduce the following notations:In these notations, the covariant version of Euler’s relation, (11), and the Gibbs-Duhem relation, (12), can be postulated asrespectively. The above equations can then be used to derive a covariant form of the first law of thermodynamics, (13):

The covariant thermodynamic relations were constructed in such a way that when (30), (31), and (32) are contracted with , we obtain the usual thermodynamic relations, (11), (12), and (13). Here we have used the property of the fluid four-velocity, . The projection of (30), (31), and (32) on the three-space orthogonal to just leads to trivial identities:From the above equations, we conclude that the covariant thermodynamic relations do not contain more information than the usual thermodynamic relations.

The first law of thermodynamics, (32), leads to the following expression for the entropy four-current divergence:After employing the conservation of energy-momentum and net particle number, (24), the above equation leads to the conservation of entropy, . It is important to note that, within equilibrium thermodynamics, the entropy conservation is a natural consequence of energy-momentum and particle number conservation and the first law of thermodynamics. The equation of motion for the entropy density is then obtained asWe observe that the rate equation of the entropy density in the above equation is identical to that of the net particle number, (28). Therefore, we conclude that, for ideal hydrodynamics, the ratio of entropy density to number density () is a constant of motion.

2.4. Relativistic Dissipative Fluid Dynamics

The derivation of relativistic ideal fluid dynamics proceeds by employing the properties of the Lorentz transformation and the conservation laws and, most importantly, by imposing local thermodynamic equilibrium. It is important to note that while the properties of Lorentz transformation and the conservation laws are robust, the assumption of local thermodynamic equilibrium is a strong restriction. The deviation from local thermodynamic equilibrium results in dissipative effects, and as all fluids are dissipative in nature due to the uncertainty principle [33], the assumption of local thermodynamic equilibrium is never strictly realised in practice. In the following, we consider a more general theory of fluid dynamics which attempts to take into account the dissipative processes that must happen, because a fluid can never maintain exact local thermodynamic equilibrium throughout its dynamical evolution.

Dissipative effects in a fluid originate from irreversible thermodynamic processes that occur during the motion of the fluid. In general, each fluid element may not be in equilibrium with the whole fluid, and, in order to approach equilibrium, it exchanges heat with its surroundings. Moreover, the fluid elements are in relative motion and can also dissipate energy by friction. All these processes must be included in order to obtain a reasonable description of a relativistic fluid.

The earliest covariant formulation of dissipative fluid dynamics was due to Eckart [34], in 1940, and, later, by Landau and Lifshitz [35], in 1987. The formulation of these theories, collectively known as first-order theories (order of gradients), was based on a covariant generalization of the Navier-Stokes theory. The Navier-Stokes theory, at that time, had already become a successful theory of dissipative fluid dynamics. It was employed efficiently to describe a wide variety of nonrelativistic fluids from weakly coupled gases such as air to strongly coupled fluids such as water. Hence, a relativistic generalization of Navier-Stokes theory was considered to be the most effective and promising way to describe relativistic dissipative fluids.

The formulation of relativistic dissipative hydrodynamics turned out to be more subtle since the relativistic generalization of Navier-Stokes theory is intrinsically unstable [36–38]. The source of such instability is attributed to the inherent acausal behaviour of this theory [46, 47]. A straightforward relativistic generalization of Navier-Stokes theory allows signals to propagate with infinite speed in a medium. While in nonrelativistic theories this does not give rise to an intrinsic problem and can be ignored, in relativistic systems, where causality is a physical property that is naturally preserved, this feature leads to intrinsically unstable equations of motion. Nevertheless, it is instructive to review the first-order theories as they are an important initial step to illustrate the basic features of relativistic dissipative fluid dynamics.

As in the case of ideal fluids, the basic equations governing the motion of dissipative fluids are also obtained from the conservation laws of energy-momentum and (net) particle number:However, for dissipative fluids, the energy-momentum tensor is no longer diagonal and isotropic in the local rest frame. Moreover, due to diffusion, the particle flow is expected to appear in the local rest frame of the fluid element. To account for these effects, dissipative currents and are added to the previously derived ideal currents, and :where is required to be symmetric in order to satisfy angular momentum conservation. The main objective then becomes to find the dynamical or constitutive equations satisfied by these dissipative currents.

2.4.1. Matching Conditions

The introduction of the dissipative currents causes the equilibrium variables to be ill-defined, since the fluid can no longer be considered to be in local thermodynamic equilibrium. Hence, in a dissipative fluid, the thermodynamic variables can only be defined in terms of an artificial equilibrium state, constructed such that the thermodynamic relations are valid as if the fluid was in local thermodynamic equilibrium. The first step to construct such an equilibrium state is to define and as the total energy and particle density in the local rest frame of the fluid, respectively. This is guaranteed by imposing the so-called matching or fitting conditions [39]:These matching conditions enforce the following constraints on the dissipative currents:Subsequently, using and , an artificial equilibrium state can be constructed with the help of the equation of state. It is, however, important to note that while the energy and particle densities are physically defined, all the other thermodynamic quantities are defined only in terms of an artificial equilibrium state and do not necessarily retain their usual physical meaning.

2.4.2. Tensor Decompositions of Dissipative Quantities

To proceed further, it is convenient to decompose in terms of its irreducible components, that is, a scalar, a four-vector, and a traceless and symmetric second-rank tensor. Moreover, this tensor decomposition must be consistent with the matching or orthogonality condition, (40), satisfied by . To this end, we introduce another projection operator, the double symmetric, traceless projector orthogonal to ,with the following properties:The parentheses in the above equation denote symmetrization of the Lorentz indices; that is, . The dissipative current then can be tensor decomposed in its irreducible form by using , , and aswhere we have definedThe scalar is the bulk viscous pressure, the vector is the energy diffusion four-current, and the second-rank tensor is the shear stress tensor. The properties of the projection operators and imply that both and are orthogonal to and, additionally, is traceless. Armed with these definitions, all the irreducible hydrodynamic fields are expressed in terms of and aswhere the angular bracket notations are defined as and .

We observe that is a symmetric second-rank tensor with ten independent components and is a four-vector; overall they have fourteen independent components. Next we count the number of independent components in the tensor decompositions of and . Since and are orthogonal to , they can have only three independent components each. The shear stress tensor is symmetric, traceless, and orthogonal to and hence can have only five independent components. Together with , , , and , which have in total six independent components ( is related to via equation of state), we count a total of seventeen independent components, three more than expected. The reason is that, so far, the velocity field was introduced as a general normalized four-vector and was not specified. Hence has to be defined to reduce the number of independent components to the correct value.

2.4.3. Definition of the Velocity Field

In the process of formulating the theory of dissipative fluid dynamics, the next important step is to fix . In the case of ideal fluids, the local rest frame was defined as the frame in which there is, simultaneously, no net energy and particle flow. While the definition of local rest frame was unambiguous for ideal fluids, this definition is no longer possible in the case of dissipative fluids due to the presence of both energy and particle diffusion. From a mathematical perspective, the fluid velocity can be defined in numerous ways. However, from the physical perspective, there are two natural choices: the Eckart definition [34], in which the velocity is defined by the flow of particles,and the Landau definition [35], in which the velocity is specified by the flow of the total energy,

We note that the above two definitions of impose different constraints on the dissipative currents. In the Eckart definition, the particle diffusion is always set to zero, while in the Landau definition, the energy diffusion is zero. In other words, the Eckart definition of the velocity field eliminates any diffusion of particles, whereas the Landau definition eliminates any diffusion of energy. In this review, we shall always use the Landau definition, (47). The conserved currents in this frame take the following form:

As done for ideal fluids, the energy-momentum conservation equation in (37) is decomposed parallel and orthogonal to . Using (48) together with the conservation law for particle number in (37) leads to the equations of motion for dissipative fluids. For , , and , one obtainsrespectively. Here , and the shear tensor .

We observe that while there are fourteen total independent components of and , (49)–(51) constitute only five equations. Therefore, in order to derive the complete set of equations for dissipative fluid dynamics, one still has to obtain the additional nine equations of motion that will close (49)–(51). Eventually, this corresponds to finding the closed dynamical or constitutive relations satisfied by the dissipative tensors , , and .

2.4.4. Relativistic Navier-Stokes Theory

In the presence of dissipative currents, the entropy is no longer a conserved quantity; that is, . Since the form of the entropy four-current for a dissipative fluid is not known a priori, it is not trivial to obtain its equation. We proceed by recalling the form of the entropy four-current for ideal fluids, (30), and extending it for dissipative fluids:The above extension remains valid because an artificial equilibrium state was constructed using the matching conditions to satisfy the thermodynamic relations as if in equilibrium. This was the key step proposed by Eckart, Landau, and Lifshitz in order to derive the relativistic Navier-Stokes theory [34, 35]. The next step is to calculate the entropy generation, , in dissipative fluids. To this end, we substitute the form of and for dissipative fluids from (48) in (52). Taking the divergence and using (49)–(51), we obtain

The relativistic Navier-Stokes theory can then be obtained by applying the second law of thermodynamics to each fluid element; that is, by requiring that the entropy production must always be positive,The above inequality can be satisfied for all possible fluid configurations if one assumes that the bulk viscous pressure , the particle-diffusion four-current , and the shear stress tensor are linearly proportional to , , and , respectively. This leads towhere the proportionality coefficients , , and refer to the bulk viscosity, the particle diffusion, and the shear viscosity, respectively. Substituting the above equation in (53), we observe that the source term for entropy production becomes a quadratic function of the dissipative currents:In the above equation, since is orthogonal to the timelike four-vector , it is spacelike and hence . Moreover, is symmetric in its Lorentz indices and in the local rest frame . Since the trace of the square of a symmetric matrix is always positive, . Hence, as long as , the entropy production is always positive. Constitutive relations for the dissipative quantities, (55), along with (49)–(51) are known as the relativistic Navier-Stokes equations.

The relativistic Navier-Stokes theory in this form was obtained originally by Landau and Lifshitz [35]. A similar theory was derived independently by Eckart, using a different definition of the fluid four-velocity [34]. However, as already mentioned, the Navier-Stokes theory is acausal and, consequently, unstable. The source of the acausality can be understood from the constitutive relations satisfied by the dissipative currents, (55). The linear relations between dissipative currents and gradients of the primary fluid dynamical variables imply that any inhomogeneity of and immediately results in dissipative currents. This instantaneous effect is not allowed in a relativistic theory which eventually causes the theory to be unstable. Several theories have been developed to incorporate dissipative effects in fluid dynamics without violating causality: Grad-Israel-Stewart theory [39, 45, 48], the divergence-type theory [49, 50], extended irreversible thermodynamics [51], Carter’s theory [52], Grmela and Ottinger theory [53], among others. Israel and Stewart’s formulation of causal relativistic dissipative fluid dynamics is the most popular and widely used; in the following we briefly review their approach.

2.4.5. Causal Fluid Dynamics: Israel-Stewart Theory

The main idea behind the Israel-Stewart formulation was to apply the second law of thermodynamics to a more general expression of the nonequilibrium entropy four-current [39, 45, 48]. In equilibrium, the entropy four-current was expressed exactly in terms of the primary fluid dynamical variables, (30). Strictly speaking, the nonequilibrium entropy four-current should depend on a larger number of independent dynamical variables, and a direct extension of (30) to (52) is, in fact, incomplete. A more realistic description of the entropy four-current can be obtained by considering it to be a function not only of the primary fluid dynamical variables but also of the dissipative currents. The most general off-equilibrium entropy four-current is then given bywhere is a function of deviations from local equilibrium, , and . Using (48) and Taylor-expanding to second order in dissipative fluxes, we obtainwhere denotes third-order terms in the dissipative currents and , and are the thermodynamic coefficients of the Taylor expansion and are complicated functions of the temperature and chemical potential.

We observe that the existence of second-order contributions to the entropy four-current in (58) should lead to constitutive relations for the dissipative quantities which are different from relativistic Navier-Stokes theory obtained previously by employing the second law of thermodynamics. The relativistic Navier-Stokes theory can then be understood to be valid only up to first order in the dissipative currents (hence also called first-order theory). Next, we recalculate the entropy production, , using the more general entropy four-current given in (58):As argued before, the only way to explicitly satisfy the second law of thermodynamics is to ensure that the entropy production is a positive definite quadratic function of the dissipative currents.