1. Introduction
Relativistic hydrodynamics has been widely applied in recent years in heavy-ion physics [
1,
2,
3,
4] and astrophysics [
5,
6,
7]. For example, the experiments at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) motivated significantly the development of the relativistic hydrodynamics with the data being well-described in terms of a fluid with low shear viscosity; for reviews, see [
8,
9]. Relativistic hydrodynamics is also used in the general relativistic simulations of compact stars in isolation or binaries, particularly in the context of gravitational wave emission by neutron star mergers observed in 2017 [
10]. A further astrophysical area of applications of relativistic hydrodynamics is the physics of compact star rotational dynamics, specifically glitches and their relaxations; for reviews, see [
11,
12].
The hydrodynamic description of fluids is valid close to the local thermal equilibrium. The hydrodynamic state of a relativistic fluid is described through its energy–momentum tensor and currents of conserved charges, which in the low-frequency and long-wavelength can be Taylor-expanded around their equilibrium values in thermodynamic gradients (so-called thermodynamic forces). The validity of such gradient expansion is guaranteed due to the clear separation between the typical microscopic and macroscopic scales of the system. The zeroth-order term in this expansion corresponds to the limit of the ideal hydrodynamics.
The truncation of the gradient expansion at the first order leads to the relativistic Navier–Stokes (NS) theory, which was worked out by Eckart [
13] and Landau–Lifshitz [
14]. It is known that the solutions of the relativistic NS equations are acausal and unstable [
15,
16,
17,
18]. The reason for the acausality is the parabolic structure of NS equations, which originated from the linear constitutive relations between the dissipative fluxes and the thermodynamic forces. Recent work demonstrated that the acausalities and instabilities in relativistic hydrodynamics are a consequence of the matching procedure to the local equilibrium reference state. More general matching conditions were used to render the theory causal and stable at first order [
19,
20,
21].
The problem of acausality can be solved in the second-order theory, where additional terms appear that contain higher (second)-order derivatives in thermodynamic quantities. For non-relativistic fluids, the second-order theory was proposed by Müller [
22], and then rediscovered and extended to relativistic systems by Israel and Stewart [
23,
24]. In these theories, the dissipative fluxes are treated as independent state variables that satisfy relaxation-type equations derivable from the entropy principle. The relaxation terms which appear in these equations recover the causality of the theory [
18,
25]. The relaxation equations for dissipative fluxes in the second-order theories and their numerical studies in simulations have been discussed extensively in the literature; see [
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37].
The relativistic second-order hydrodynamics can be obtained from moments of the Boltzmann equation for the distribution function [
38,
39,
40]. This theory provides a systematic way of evaluating the new coefficients describing the relaxation of dissipative quantities at weak coupling and in the quasiparticle limit. An alternative approach valid in the strong coupling limit is Zubarev’s non-equilibrium statistical operator formalism, which was recently applied to obtain the second-order hydrodynamics and/or Kubo-type formulae for transport coefficients [
41,
42,
43]. Related approaches based on field theory methods are given in Refs. [
44,
45].
Recent applications of relativistic hydrodynamics focused on the systems featuring spin, polarization, and vorticity. A systematic calculation of the corrections of the stress–energy tensor and currents up to the second order in thermal vorticity were given in Refs. [
46,
47] and the relevant Kubo formulae were derived. Similarly, the problem of relativistic hydrodynamics under strong spin was studied on the basis of entropy–current analysis in Ref. [
48] where the seventeen transport coefficients of highly anisotropic relativistic hydrodynamics were identified. Ref. [
49] formulated relativistic spin hydrodynamics of Dirac fermions in a torsionful curved background and derived the relevant Kubo formulae associated with the correlation functions. The spin hydrodynamics was also derived from the Boltzmann equation using the method of moments in Ref. [
50] up to the second order. Plasmas with vorticity were studied in second-order dissipative hydrodynamics in the presence of a chiral imbalance in Ref. [
51] and the dispersion relations of chiral vortical waves were obtained.
Specific applications of second-order hydrodynamics include, for example, the computation of the transport coefficients from a Chapman–Enskog-like expansion for a system of massless quarks and gluons [
34]. The second-order hydrodynamics was recently applied to multicomponent systems with hard-sphere interactions in Ref. [
52], where the multicomponent nature of the system was taken into account. The second-order hydrodynamics was applied to quark–gluon plasma in heavy ion collisions, for example, in Ref. [
53], showing the coupling between diffusion and shear and bulk viscosities in a multi-component system. A discussion of the use of relativistic anisotropic hydrodynamics to study the physics of ultrarelativistic heavy-ion collisions was given in Ref. [
54], using as the main ingredient the quasiparticle anisotropic hydrodynamics model for quark–gluon plasma. Anomalies in the field theory were included in the description of heavy-ion collisions in Ref. [
55]. The strong gravity regime of second-order hydrodynamics, relevant for astrophysical applications, was studied in Ref. [
56]. The first-order theory was applied in the context of the out-of-equilibrium dynamics of viscous fluids in a spatially flat cosmology [
57].
Recent work also extended the current formulations of relativistic hydrodynamics to include the effect of magnetic fields. The first-order dissipative effects and relevant transport coefficients were derived, and a numerical method was developed using the method of moments in Ref. [
58]. The effects of anomalies in the magnetohydrodynamics were included in Ref. [
59]. Charge diffusion in the second-order theories was recently studied in Ref. [
60]. Applications to neutron stars and relevant transport coefficients were derived, for example, in Refs. [
61,
62,
63]. A comprehensive review of the new developments in this field is given in Ref. [
64].
More formal recent developments of relativistic hydrodynamics include formulations that avoid specific frames. (The choice of frames in relativistic heavy ion collisions was discussed in Ref. [
65].) First- and second-order viscous relativistic hydrodynamics were formulated without specific frame conditions in Refs. [
21,
66], respectively, with the causality and stability conditions explicitly formulated in the conformal regime in the second work. Earlier, the stability of first-order hydrodynamics was established in Ref. [
20].
The purpose of this article is twofold. First, we review the phenomenological derivation of the second-order dissipative hydrodynamics from general principles of conservation laws and the second law of thermodynamics. The dissipative equations are obtained from expansions of the energy–momentum tensor and currents around their equilibrium values in thermodynamic gradients (so-called thermodynamic forces) order by order. Second, we consider the theory in the general case of a system with
l independent flavors of conserved charges, keeping all possible second-order terms which arise from the entropy current and the second law of thermodynamics. In the case where the nonlinear terms in the thermodynamic gradients and dissipative fluxes are dropped, these equations are reduced to the original Israel and Stewart hydrodynamics [
23,
24].
This work is organized as follows. In
Section 2, we present the ideal hydrodynamics of non-dissipative fluids. Dissipation in the fluids is introduced in
Section 3. We discuss the first-order hydrodynamics and obtain the Navier–Stokes equations in
Section 4. The second-order Israel–Stewart theory in the case of a system with
l independent flavors of conserved charges is given in
Section 5. A summary is provided in
Section 6.
2. Relativistic Ideal Hydrodynamics
Relativistic hydrodynamics describes the state of the fluid employing its energy–momentum tensor
and currents of conserved charges
, such as the baryonic, electric, etc. Here, we consider the general case of a system with
l independent flavors of conserved charges, which are labeled by the index
a. The equations of relativistic hydrodynamics are contained in the conservation laws for the energy–momentum tensor and the charge currents
In dissipative hydrodynamics, also the entropy principle
should be applied to close the system (
1), where
is the entropy 4-current.
Ideal hydrodynamics corresponds to the zeroth-order expansion of the energy-momentum tensor and charge currents with respect to the thermodynamic forces. In this case, each fluid element maintains the local thermal equilibrium during its evolution [
14,
67]. The macroscopic state of the fluid is therefore fully described through the fields of the energy density
and the charge densities
. The fluid 4-velocity is defined as
where
is the displacement 4-vector of a fluid element,
is the infinitesimal change in the proper time,
is the fluid 3-velocity and
is the relevant Lorentz factor. The 4-velocity given by Equation (
2) is normalized by the condition
and has only three independent components.
Because the thermal equilibrium is maintained locally, each fluid element can be assigned well-defined local values of the temperature
, chemical potentials
(conjugate to charge densities
), entropy density
and pressure
. These quantities are related to the local energy and charge densities via an equation of state
and the standard thermodynamic relations
where
h is the enthalpy density.
The fluid 4-velocity is defined in such a way that in the fluid rest frame, the energy and charge flows vanish:
and
, if
. These conditions together with the spatial isotropy imply the following form for the energy–momentum tensor and charge currents [
14,
67]:
where the index 0 labels the quantities in ideal hydrodynamics;
is the projection operator onto the 3-space orthogonal to
and has the properties
From Equations (
6) and (
7), we obtain
In the fluid rest frame,
; therefore,
and
. In this case, Equation (
8) simplifies to
We introduce also the energy 4-current or momentum density by the formula
which is the relativistic generalization of the 3-momentum density (mass current). As seen from Equations (
6) and (
10), all charge currents are parallel to each other and to the energy flow, which is due to the possibility of a unique definition of the velocity field for ideal fluids.
The equations of ideal hydrodynamics are obtained by substituting the expressions (
6) into the conservation laws (
1)
Contracting the second equation in (
11) once with
and then with the projector
and taking into account Equation (
7) and the condition
, we obtain
where we introduce the covariant time derivative, covariant spatial derivative, and velocity 4-divergence via
,
and
, respectively. The velocity 4-divergence
quantifies how fast the fluid is expanding (
) or contracting (
); it is called the
fluid expansion rate. In the case of an incompressible flow of the fluid, we have
.
It is not difficult to recognize in the first two relations in (
12) the covariant expressions for the charge conservation law and the energy conservation law, respectively. The third equation is nothing more than the relativistic generalization of the ordinary Euler equation familiar from nonrelativistic hydrodynamics. From these equations, one can deduce that in relativistic hydrodynamics, the role of rest mass density is taken over by the enthalpy density
h, which thus provides the correct inertia measure for relativistic fluids. This fact illustrates the importance of the quantity (
10) as the relativistic analog of the momentum flux.
The system (
12) contains
equations for
variables
,
p,
and
. To close the system, one still needs to specify an equation of state
, which relates the pressure to the conserved thermodynamic variables.
It is easy to show that the equations of ideal hydrodynamics lead automatically to entropy conservation. The entropy flux can be written as
Using the thermodynamic relations (
3)–(
5) and the equations of motion (
12), we obtain
which is the second law of thermodynamics for a non-dissipative system. Using Equations (
5), (
6), (
10) and (
13), we can rewrite the entropy current as
where we defined
The expression (
15) is the covariant form of the relation (
5). To proceed further, it is convenient to modify Equations (
3) and (
4) using the definitions (
16). We obtain for Equation (
4)
where we used Equation (
5) in the second step. Now the first law of thermodynamics and the Gibbs–Duhem relation can be written in an alternative form:
With the aid of Equations (
6) and (
15), these relations can be cast into a covariant form:
where we used the second relation in Equation (
18). One should note that, despite their vector form, these equations do not contain more information than the scalar thermodynamic relations (contraction of Equations (
19) and (
20) with the projector
leads to identities).
4. Relativistic Navier-Stokes (First-Order) Theory
The entropy conservation law (
14) derived in the framework of ideal fluid dynamics is no longer valid for dissipative fluids. In this case, it should be replaced by the second law of thermodynamics, which implies that the entropy production rate of an isolated system must be always non-negative:
where equality holds only for reversible processes. By analogy with the decomposition of the charge currents (
31), we can decompose
into contributions parallel and orthogonal to
where
s is identified with the equilibrium entropy density (which is sufficient for the first-order accuracy, as explained in
Section 3.1), and the vector
satisfies the condition
. The index 1 denotes that (
58) is only the first-order approximation to the entropy flux. For small departures from equilibrium, it is natural to assume that
is a linear combination of the energy and charge diffusion fluxes
where
and
are functions of thermodynamic variables and should be determined from the condition (
57). This formulation of relativistic dissipative fluid dynamics was proposed by Eckart [
13] and Landau–Lifshitz [
14], and leads to the relativistic version of the NS theory. Inserting Equation (
59) into Equation (
58) and introducing the dissipation function via
, we obtain
where we employed the relations (
3) and (
5) and eliminated the terms
and
using Equations (
54) and (
55). Requiring
, we identify from Equation (
60)
,
. In the third and the fourth terms, we can replace
due to the orthogonality conditions (
33). Then, we have
We can further simplify the third term by approximating
from Equation (
12) since
is already of the order
. Using Equation (
18), we obtain
Substituting this result into Equation (
61) and recalling the definition (
44), we obtain finally
The second law of thermodynamics implies
, which requires the right-hand side of Equation (
63) to be a quadratic form of thermodynamic forces
,
and
. Assuming linear dependence of the dissipative fluxes on the thermodynamic forces, we obtain the constitutive relations
where
and
are called the shear and the bulk viscosities, respectively, and
is the matrix of the diffusion coefficients. These relations together with Equations (
54)–(
56) constitute a closed system of equations, which are known as relativistic NS equations.
From Equations (
63) and (
64), we obtain for the dissipation function
The positivity of this expression is guaranteed by the positivity of the viscosity coefficients and the eigenvalues of the matrix
(recall that the diffusion fluxes
are spatial). We see from Equation (
65) that the contribution of diffusion processes to the function
R depends only on the relative diffusion currents
, but not on the currents
and
separately. This result, which was obtained in the framework of the relativistic NS (first-order) theory, is the direct consequence of the Lorentz invariance and indicates that the dissipation in the fluid is independent of the choice of the fluid velocity field, as expected.
The entropy current given by Equations (
58) and (
59) can now be written as
where we used Equations (
5), (
16), (
32) and (
44) to obtain the second relation. One can give a simple interpretation of the second expression in Equation (
66). Recalling that
is the fluid velocity measured in the L-frame (see Equation (
32)), we observe that the first term on the right-hand side of Equation (
66) is the entropy current that is convected together with the energy. The second term arises as a result of the relative flow between the energy and the charges and, therefore, should be identified with the irreversible part of the entropy flow. Using Equations (
5), (
16), (
32) and (
44), we can rewrite Equation (
66) also in the following form:
which formally coincides with its counterpart of ideal hydrodynamics (
15).
If we have only one sort of conserved charge (
), then instead of the third relation in Equation (
64), we have simply
where we used the definition of the heat current given by Equation (
51) and introduced the coefficient of thermal conductivity via
Equation (
69) establishes the relation between the thermal conductivity and the diffusion coefficient
. Thus, there are only three independent transport coefficients in the first-order theory which relate the irreversible fluxes to the corresponding thermodynamic forces. Employing Equation (
62), we can write the heat flux (
68) in the following way
which in the fluid rest frame reads
Equation (
71) is the relativistic generalization of the well-known Fourier law
of the non-relativistic hydrodynamics [
14].
The expressions (
65)–(
67) in the case of
reduce to
and
The last relation in Equation (
73) is the decomposition of the entropy flow into its reversible and irreversible components observed from the E-frame.
Note that in the case where there are no conserved charges, i.e., when
, the heat conduction and/or diffusion phenomena are absent [
69].
5. Israel–Stewart (Second-Order) Theory
The first-order theory described in the previous section turns out to be acausal, and, therefore, cannot be regarded as a consistent theory of relativistic dissipative fluids. The origin of acausality lies in the constitutive relations (
64), which imply that the thermodynamic forces generate dissipative fluxes instantaneously [
15,
17,
22,
23]. In addition, this theory suffers also from instability, which is a consequence of acausality [
15,
16,
17,
18,
25]. However, as pointed out in the introduction, acausalities, and instabilities are a consequence of the matching procedure to the local-equilibrium reference state which can be generalized to obtain causal and stable first-order dissipative hydrodynamics [
19,
20,
21].
It turns out that to recover the causality, the entropy current
is required to be at least a quadratic function of the dissipative fluxes. This idea of an extension of the entropy current up to the second order was first proposed by Müller [
22] for nonrelativistic fluids. For relativistic fluids, a similar second-order theory was developed by Israel and Stewart [
23,
24]. In this subsection, we review briefly the Israel–Stewart (IS) formulation of causal hydrodynamics, following mainly Ref. [
23].
As a starting point, the entropy current given by Equations (
66) and (
67) is extended up to the second order in dissipative quantities
,
,
and
. It is worth stressing that, despite the frame dependence of these quantities, the entropy current
along with the energy–momentum tensor
and the charge currents
should be regarded as a
primary variable and should be therefore frame-independent up to the second order [
23,
24] (we note that the expressions (
66) and (
67) are frame-independent only at the first order in deviations from equilibrium).
We write now the entropy current in the following form:
where
is the first-order contribution given by Equations (
66) and (
67), and the terms
and
collect all possible second-order corrections.
The most general form of the vectors
and
in a generic frame is [
23]
where the new coefficients
,
,
,
and
are unknown functions of
and
. The vector
is frame-independent up to the second order, and
collects the second-order contributions to
which are not frame-independent to the order
(note that we use different metric signature from Ref. [
23]). Because the contribution
is frame-independent only to the first order, the term
is necessary to provide the frame-independence of the total entropy current (
74) [
23].
The terms in Equations (
75) and (
76) which are proportional to
are responsible for the second-order corrections to the equilibrium entropy density
s. Indeed, the non-equilibrium entropy density is identified with
, which can be found from Equations (
74)–(
76)
where we used Equations (
66). The last inequality in Equation (
77) requires the entropy density in non-equilibrium states to be smaller than its equilibrium value
s. From here, we conclude that
,
, and the matrix
is positive-semidefinite. The term in Equation (
77), which is proportional to
represents the shift in the entropy density because of the change of the reference frame and is automatically negative. Those terms in Equations (
75) and (
76) which are orthogonal to
represent the irreversible entropy flux arising from couplings between the diffusion and viscous fluxes.
The phenomenological equations for the dissipative fluxes should be found again from the positivity condition of the dissipative function
The first term in Equation (
78) was already computed in Equation (
61):
Now we use Equation (
56) to eliminate the acceleration term in Equation (
79):
where we used the second relation of Equation (
18) to modify the pressure gradient in Equation (
80). Equation (
80) differs from Equation (
62) by additional second-order terms, which now cannot be neglected. Substituting Equation (
80) in the third term of Equation (
79) and recalling the definitions (
44), we obtain
where we used the notation
. Using Equations (
76) and (
81), we obtain
Here, we introduced three additional coefficients
,
and
because there is an ambiguity in “sharing” the terms involving
and
between the diffusion and viscous fluxes [
15]. Note that Ref. [
23] assumed
.
Now we take the divergence of Equation (
75):
where we introduced again two additional coefficients
and
to “share” the terms containing
and
between the diffusion and viscous fluxes [
15]. We kept also all terms that contain gradients of transport coefficients which were neglected in Ref. [
23].
Combining Equations (
82) and (
83), we obtain for the dissipative function (
78)
where we introduced the short-hand notations
and used the symmetries of the dissipative fluxes.
The expressions in the square brackets in Equation (
84) are called
generalized or extended thermodynamic forces. Requiring
, fixing for simplicity the L-frame
and assuming linear relations between these forces and the dissipative fluxes, we obtain the following evolution equations:
Equations (
87)–(
89) in this form for
single type of conserved charges were derived in Ref. [
15] and were written in the E-frame. In the original papers of Israel and Stewart [
23,
24,
70], the terms which are nonlinear in thermodynamic forces and dissipative fluxes were dropped, although the general case of chemically reacting multicomponent mixtures was discussed in Ref. [
23]. The current form generalizes the hydrodynamics equations derived in Refs. [
15,
23,
24,
70] to the case of multiple independent flavors of conserved charges, where all second-order terms arising from the entropy current are kept.
With the aid of Equations (
87)–(
89), the dissipative function (
84) obtains the form
which formally coincides with Equation (
65). Defining relaxation times according to
we can write Equations (
87)–(
89) in the following form:
The first terms on the right-hand sides of these equations represent the corresponding NS contributions to the dissipative fluxes; see Equation (
64). The first terms on the left-hand sides incorporate the relaxation of the dissipative fluxes to their NS values on finite time scales given by Equation (
91). Thus, these relaxation terms imply a delay in the response of the dissipative fluxes to thermodynamic forces and recover the causality of the theory [
18,
25]. The rest of the terms in Equations (
92)–(
94) are responsible for spatial inhomogeneities in the dissipative fluxes as well as nonlinear couplings between different dissipative processes.
We note that the derivation of the second-order hydrodynamics from the kinetic theory produces additional terms which are not obtained within the phenomenological theory [
24]. The derivation of complete IS equations from kinetic theory is discussed in Refs. [
30,
38,
71,
72,
73].
In the case of one conserved current, we have instead of Equations (
92)–(
94)
where
.
6. Summary
We provided a review of the phenomenological theory of second-order relativistic hydrodynamics for systems with multiple conserved charges with an extension to multiflavor fluids. The hydrodynamic state of the system is described through the energy–momentum tensor and the 4-currents of conserved charges. We reviewed the derivation and content of the equations at zeroth, first, and second order in gradient expansion of the energy–momentum tensor and currents, which led us to the ideal Navier–Stokes and Israel–Stewart hydrodynamics, respectively. From the positivity condition of the dissipative function, the most general set of dissipative processes was identified which contains also the relative diffusions between different conserved currents. We kept all the second-order gradient terms arising from the second law of thermodynamics, as well as the nonlinear terms in the thermodynamic gradients and dissipative fluxes, which were omitted in the canonical Israel–Stewart theory, but were kept in Ref. [
15] in the case of one conserved flavor.
The hydrodynamics theory exposed in this article is phenomenological in nature. More formal but also complex derivations are available in the literature which utilize different concepts and approaches of statistical mechanics, for example, quasiparticle Boltzmann equation [
38,
39,
40] or non-equilibrium statistical operator [
41,
43]. Nevertheless, the phenomenological theory can be reasonably expected to remain in the arsenal of theoretical tools that can be deployed in studies of fluid in new settings and/or under various external fields.
A few closing remarks are in order concerning the numerical implementations of the second-order dissipative hydrodynamics. First of all, the choice between the frames (i.e., Eckart vs. Landau) may be significantly influenced by the required computational cost, which in turn depends on such factors as system size, boundary conditions and the level of accuracy required. In general, the Landau frame is known to be computationally more expensive compared to the Eckart frame, but it also provides more insight into the behavior of the fluid. Eventually, the choice between the Landau and Eckart frames will depend on the specific problem at hand and the trade-off between accuracy and computational efficiency. A number of methods are available for the solution of the generalized (second-order) Navier–Stokes equations for multifluids, for example, finite difference method, finite volume method, spectral methods, and Lagrangian particle methods; see Ref. [
9]
Section 5 for a pedagogical discussion in the context of relativistic heavy ion collisions. Again, the choice of the method will be dictated by the specifics of the problem at hand, the computational cost, and required accuracy.