Mapping GENERIC Hydrodynamics into Carter’s Multifluid Theory

Abstract

We show that the GENERIC model for relativistic heat conduction is a multifluid of Carter; this allows one to compute the multifluid constitutive relations directly from the GENERIC formalism. As a quick application, we prove that in the limit of infinite heat conductivity, GENERIC heat conduction reduces to the relativistic two-fluid model for superfluidity. This surprising “crossover” is a consequence of relativistic causality: if diffusion happens too fast, all the diffusing charge cumulates on the surface of the light cone, and it eventually travels at the speed of light like a wave. Our analysis is non-perturbative and carried out in a fully non-linear regime.

1. Introduction

Most of the relativistic hydrodynamic theories currently in use rely on some form of near-equilibrium expansion. For example, the original Israel–Stewart theory arises from a perturbative expansion for small viscous fluxes [1,2,3,4], while BRSSS [5] and BDNK [6] arise from an expansion for small gradients. DNMR arises from a subtle combination of the two expansions above [7]. While these approaches have the advantage of being systematic and rigorous, the resulting fluid models usually have a limited formal regime of applicability, and they may not capture all those intrinsically non-linear phenomena that real-world substances are known to exhibit. This kind of problem is well known to material engineers since, in our everyday life, we deal with many non-Newtonian fluids that easily escape a Navier–Stokes description [8]. Indeed, most foods and toiletry products explore dynamical regimes where the shear stress component, say, ${\Pi }_{12}$, grows like a fractional power of the symmetrized velocity gradient ${\sigma }_{12}={\partial }_{(1}{v}_{2)}$ [9]. For example, the shear stress of ketchup at 30.5 ${}^{°}$C exhibits the following non-Newtonian behavior:

$${\Pi }_{12}\propto |{\sigma }_{12}{|}^{0.136}\mathrm{for}\left|{\sigma }_{12}\right|\in [0.1,100]{\mathrm{s}}^{-1}.$$

(1)

Clearly, features of this kind are hardly captured by conventional gradient expansions.

Since the rheological properties of hadronic and quark matter are poorly known, we cannot exclude that non-perturbative corrections to the constitutive relations might be important for neutron star mergers and heavy-ion phenomenology. If that were the case, relativistic viscous hydrodynamics as we know it would be non-reliable for most practical applications, and we would need a full-fledged theory of relativistic rheology. At present, a fully general and rigorous theory of this kind exists only in the linear regime [10], and we are still unable to describe non-perturbative phenomena systematically. Still, some isolated “non-linear rheological models” have been formulated, such as Anisotropic Hydrodynamics [11,12], which models shear viscosity non-perturbatively, and Hydro+ [13,14], which models bulk viscosity non-perturbatively. The question of how to model non-perturbative corrections to heat conduction is still an open problem, and it is the subject of investigation in the present article.

First, a bit of history. The interplay between heat conduction and relativity has had a long (and controversial) development. The source of all problems is a technicality: if $E=m{c}^2$, then “flow of energy” = “density of momentum” (i.e., $T^{j0}=T^{0j}$ [15]). As a consequence, heat carries inertia, and it is practically impossible to disentangle exchanges of heat from macroscopic accelerations [16]; this is a natural consequence of the relativistic duality between energy and momentum, but failure to recognize it has generated some confusion over the definition of thermal equilibrium in relativity [17]. This subtlety is also at the origin of the (unphysical) heat-driven accelerations predicted by the Eckart theory [18], and of the ambiguity between the Eckart and the Landau–Lifshitz hydrodynamic frames [4].

The modern theory of relativistic heat conduction was pioneered by Carter [19,20]. He started from the observation that if the heat carries inertia, then the four-momentum of the fluid must be “redistributed” between the entropy current $s^{\alpha }$ and the particle current $n^{\alpha }$. By symmetry, the most general way of distributing the four-momentum is

$$T^{\alpha \beta }=P{g}^{\alpha \beta }+\tilde{C}{s}^{\alpha }{s}^{\beta }+\tilde{A}(s^{\alpha }{n}^{\beta }+n^{\alpha }{s}^{\beta })+\tilde{B}{n}^{\alpha }{n}^{\beta },$$

(2)

where P, $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ are functions of $s^{\alpha }$ and $n^{\alpha }$; this gives rise to a “two-fluid model” for heat conduction, where dissipation is represented as friction between the two currents $s^{\alpha }$ and $n^{\alpha }$. Not surprisingly, one can show that when the currents are almost collinear, Carter’s theory reduces to the Israel–Stewart theory [21,22,23]. However, one can, in principle, extend the constitutive relations of P, $\tilde{A}$, $\tilde{B}$, $\tilde{C}$ to states where the angle between $s^{\alpha }$ and $n^{\alpha }$ is large, allowing one to model heat conduction in non-perturbative regimes [24].

Unfortunately, at present, the applicability of Carter’s theory to non-perturbative regimes remains merely hypothetical. In fact, there is no concrete proposal for the constitutive relations of P, $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ beyond the Israel–Stewart limit. In this brief article, we present a novel and rigorous result which might be helpful in this direction. In fact, here we prove that there is an exact change in variables that maps the GENERIC theory (GENERIC is an acronym for “General Equation for the NonEquilibrium Reversible-Irreversible Coupling” [25]. It is grounded on the assumption that the equations of motion of any non-equilibrium thermodynamic system, with degrees of freedom $\Psi$, can be expressed in the form $\dot{\Psi }=F_H(\Psi )+F_S(\Psi )$, where $F_H$ is a reversible Hamiltonian part and $F_S$ is an irreversible entropy-driven Lyapunov part) for non-linear heat conduction developed by Öttinger [26] into Carter’s multifluid theory. This implies that it is possible to derive the constitutive relations of Carter’s theory directly from the GENERIC formalism [25,27]. Since the latter was designed with the very purpose of modeling non-perturbative phenomena in complex fluids [27], this might be our only way of deriving Carter’s theory for heat conduction directly from microphysics.

Throughout this paper, we adopt the metric signature $(-,+,+,+)$, and work in natural units: $c=k_B=\hslash =1$.

2. Relativistic Dissipative Hydrodynamics in a Nutshell

Historically, there have been two different ways of modeling dissipation in a hydrodynamic setting. One is the so-called “gradient expansion”, where the degrees of freedom of the theory are the ordinary fluid fields $\{T,\mu ,u^{\alpha }\}$, representing, respectively, temperature, chemical potential, and flow velocity (each field corresponds to a conservation law, respectively energy, particles, and momentum [28,29]), and the stress–energy tensor is expanded in powers of gradients of such variables, e.g., $T^{\mu \nu }=T^{\mu \nu }\left({\partial }^0\right)+T^{\mu \nu }\left({\partial }^1\right)+T^{\mu \nu }\left({\partial }^2\right)+\mathcal{O}\left({\partial }^3\right)$. This approach goes back to Navier and Stokes [30] and Burnett [31], and it is often invoked as the main rationale for viscous hydrodynamics [5,32,33]. The second is known as “Extended Irreversible Thermodynamics” [34] (EIT), and it consists of adding new non-equilibrium variables besides the usual fluid ones, i.e., $\{T,\mu ,u^{\alpha },X,Y,Z,\dots \}$. This expands the state space of the fluid, enabling dissipation when the new variables differ from their local equilibrium values. These extended fluid variables fully characterize all the properties of the fluid at a given event $x^{\mu }$. Thus, no gradient corrections are needed, i.e., $T^{\mu \nu }=T^{\mu \nu }(T,\mu ,u^{\alpha },X,Y,Z,\dots )$. This approach was pioneered by Maxwell [35], Cattaneo [36], and Muller [37], and it finds many applications in the theory of viscoelasticity [9,38,39,40,41].

To better visualize the difference between the two approaches, consider the problem of deriving hydrodynamics from kinetic theory. In this setting, the goal is to express the one-particle distribution function $f_p\left(x^{\mu }\right)$ in terms of a set of fluid fields $\Psi \left(x^{\mu }\right)=\{{\Psi }_1\left(x^{\mu }\right),{\Psi }_2\left(x^{\mu }\right),\dots \}$, i.e., $f_p=f_p[\Psi ]$. Then, the relativistic Boltzmann equation [15,42]

$${\displaystyle \frac{d}{d\lambda }}{f}_p{|}_{\mathrm{Geodesic}\left(x\left(\lambda \right),p\left(\lambda \right)\right)}=“\mathrm{Collision}\mathrm{integral}”,$$

(3)

written in terms of $\Psi$, can be used to derive some approximate equations of motion for the fluid fields.

In the ideal fluid limit, the distribution is locally Fermi–Dirac for Fermions, or Bose–Einstein for Bosons, so that

$$f_p(T,\mu ,u^{\alpha })={\displaystyle \frac{1}{e^{-(\mu +u_{\alpha }{p}^{\alpha })/T}\pm 1}}(+\mathrm{for}\mathrm{Fermions};-\mathrm{for}\mathrm{Bosons}).$$

(4)

This clearly shows that, for ideal fluids, the natural field degrees of freedom are $\Psi =\{T,\mu ,u^{\alpha }\}$. Indeed, by substituting (4) into (3), one can easily derive the equations of motion for the ideal fluid [43].

In the dissipative case, one needs to model the non-equilibrium deviations from (4). In the gradient expansion approach, one still assumes that $\Psi =\{T,\mu ,u^{\alpha }\}$, but now $f_p$ becomes a functional of $\Psi$, i.e.,

$$f_p[T,\mu ,u^{\alpha }]={\displaystyle \frac{1}{e^{-(\mu +u_{\alpha }{p}^{\alpha }+{\varphi }_p[T,\mu ,u^{\alpha }])/T}\pm 1}}(+\mathrm{for}\mathrm{Fermions};-\mathrm{for}\mathrm{Bosons}),$$

(5)

where ${\varphi }_p[\Psi ]$ depends on the value of $\Psi$ in a (possibly large) neighborhood $\mathcal{U}\left(x^{\mu }\right)$ of the event $x^{\mu }$ under consideration. Now, if T, $\mu$, and $u^{\alpha }$ are analytic across $\mathcal{U}\left(x^{\mu }\right)$, their behavior in the neighborhood is uniquely determined by the values of all their infinite derivatives at $x^{\mu }$ [44] so that

$${\varphi }_p[\Psi ]={\varphi }_p(\Psi ,{\partial }_{\mu }\Psi ,{\partial }_{\mu }{\partial }_{\nu }\Psi ,\dots ).$$

(6)

Expanding ${\varphi }_p$ in powers of derivatives leads to viscous theories, such as relativistic Navier–Stokes [30,45], BDNK [6,46,47], BRSSS [5], and IRED [48]. Adding up all the terms of the series of ${\varphi }_p$ usually leads to a divergence. However, with the aid of Borel resummation, one can reconstruct, in some cases, the exact value of ${\varphi }_p$ [49,50,51,52].

The approach of Extended Irreversible Thermodynamics is different. Here, one parameterizes the quantity ${\varphi }_p$ in Equation (5) in terms of some structural fields $X_n\left(x^{\mu }\right)$, which determine its analytical form. For example, one may express ${\varphi }_p\left(x^{\mu }\right)$ as a series ${\varphi }_p\left(x^{\mu }\right)={\sum }_n{X}_n\left(x^{\mu }\right)g_n\left(p\right)$, where $g_n\left(p\right)$ form a basis of $L^2\left(“\mathrm{Mass}\mathrm{hyperboloid}”\right)$ [15]. This results in a theory with hydrodynamic fields $\Psi =\{T,\mu ,u^{\alpha },X_n\}$, and distribution function

$$f_p(T,\mu ,u^{\alpha },X_n)={\displaystyle \frac{1}{e^{-[\mu +u_{\alpha }{p}^{\alpha }+{\varphi }_p\left(X_n\right)]/T}\pm 1}}(+\mathrm{for}\mathrm{Fermions};-\mathrm{for}\mathrm{Bosons}).$$

(7)

No information about the local gradients enters the formula for $f_p$ (and thus that of $T^{\mu \nu }$). Instead, the gradients appear when we substitute (7) into (3). This produces some equations of motion for $X_n$, where the local gradients appear as sources. If we truncate the sequence of $X_n$ (invoking some power counting scheme) and keep only the most relevant variables, we recover transient hydrodynamic theories, such as Israel–Stewart hydrodynamics [1], DNMR [53] (in part [7]), the divergence-type theory [54], and maximum entropy hydrodynamics [55]. Carter’s theory and the GENERIC adopt the EIT approach, too. Therefore, we will stick here to interpretation (7).

We remark that one should not view (5) and (7) as contradictory to each other. Rather, they are different techniques for approximating the same complicated system. As such, they focus on different aspects of the problem. The mathematical relationship between the two viewpoints has been studied systematically in several works [56,57,58,59].

3. Outline of Carter’s Multifluid Theory

In this section, we briefly review Carter’s multifluid theory. The reader can see [19,60,61,62,63,64,65] for an overview of the foundations of the formalism. For the purposes of this paper, it will be enough to adopt the generating function approach developed in [24]. Also, we will work in the “pressure & momenta” representation [23,66].

3.1. Multifluid Constitutive Relations

Our fundamental degrees of freedom are two (unconstrained) covectors, $w_{\mu }$ and ${\xi }_{\mu }$, which are called momenta of the fluid. The number of algebraic degrees of freedom is, therefore, $8=5\left(\mathrm{equilibrium}\right)+3\left(\mathrm{heat}\mathrm{flux}\right)$, following the Israel–Stewart theory for heat conduction (Carter’s theory does not pick a specific hydrodynamic frame, i.e., it is not explicit on the definitions of T, $\mu$, and $u^{\mu }$ out of equilibrium. However, the “natural” frames take the form $u^{\mu }=a{w}^{\mu }+b{\xi }^{\mu }$, $T=-u^{\mu }{w}_{\mu }$, and $\mu =-u^{\mu }{\xi }_{\mu }$ [20,62,64]). We anticipate here that $w_{\mu }$ coincides with the homonymous structural variable introduced by Öttinger [26]. The multifluid equation of state of the system is an expression of the form

$$P=P(w_{\mu },{\xi }_{\mu },g^{\alpha \beta }),$$

(8)

where P is the kinematic pressure of the multifluid transversal to the heat flux, and $g^{\alpha \beta }$ is the inverse metric tensor. The constitutive relations for the entropy current $s^{\mu }$, the conserved particle current $n^{\mu }$, and the (symmetric) stress–energy tensor $T^{\mu \nu }$ are computed from the equation of state through the following differential:

$${\displaystyle \frac{d\left(\sqrt{-g}P\right)}{\sqrt{-g}}}=-s^{\mu }d{w}_{\mu }-n^{\mu }d{\xi }_{\mu }-{\displaystyle \frac{1}{2}}{T}_{\alpha \beta }d{g}^{\alpha \beta }.$$

(9)

Lorentz covariance implies that P can depend only on the scalars $w^{\mu }{w}_{\mu }$, $w^{\mu }{\xi }_{\mu }$, and ${\xi }^{\mu }{\xi }_{\mu }$ (recall that, within Extended Irreversible Thermodynamics, gradients do not enter the constitutive relations; see Section 2). Therefore, we can express the differential of P in the following form [66]:

$$-2dP=Cd\left(g^{\alpha \beta }{w}_{\alpha }{w}_{\beta }\right)+2Ad\left(g^{\alpha \beta }{\xi }_{\alpha }{w}_{\beta }\right)+Bd\left(g^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }\right),$$

(10)

where C, A, and B are some non-equilibrium thermodynamic scalars. Substituting (10) into (9), we obtain

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{s}^{\mu }\\ n^{\mu }\end{array}\right]=& \left[\begin{array}{cc}C& A\\ A& B\end{array}\right]\left[\begin{array}{c}{w}^{\mu }\\ {\xi }^{\mu }\end{array}\right],\hfill \\ \hfill T^{\alpha \beta }=& P{g}^{\alpha \beta }+s^{\alpha }{w}^{\beta }+n^{\alpha }{\xi }^{\beta }.\hfill \end{array}$$

(11)

Note that the stress–energy tensor above has the form (2), and it is, therefore, symmetric, even if not manifestly so. The inverse of the $2\times 2$ matrix in the first equation of (11) is called the entrainment matrix, and it enters the quadratic form in Equation (2). Both the entrainment matrix and its inverse are positive definite by stability [23]. This results in three notable inequalities: $C>0$, $B>0$, and $CB-A^2>0$.

3.2. Multifluid Field Equations

The fields of the hydrodynamic theory are ${\phi }_i=\{w_{\mu },{\xi }_{\mu }\}$, which constitute eight algebraic degrees of freedom. Hence, we need eight independent equations of motion. Out of these, 5 are the conservation laws, ${\nabla }_{\mu }{n}^{\mu }=0$, and ${\nabla }_{\mu }T{}_{\nu }^{\mu }=0$. Invoking (10) and (11), the energy–momentum conservation can be equivalently rewritten in the form [62]

$${\nabla }_{\mu }T{}_{\nu }^{\mu }=2s^{\mu }{\nabla }_{[\mu }{w}_{\nu ]}+2n^{\mu }{\nabla }_{[\mu }{\xi }_{\nu ]}+w_{\nu }{\nabla }_{\mu }{s}^{\mu }=0.$$

(12)

To complete the system, we need three additional equations of motion. The multifluid theory does not provide a unique choice for such an equation. Instead, it provides us with some reasonable options inspired by geometry and thermodynamics [19,20,24,65]. A rather convenient option is

$$2n^{\mu }{\nabla }_{[\mu }{w}_{\nu ]}=-\Xi \left(g{}_{\nu }^{\mu }-{\displaystyle \frac{n^{\mu }{n}_{\nu }}{n^{\alpha }{n}_{\alpha }}}\right)w_{\mu },$$

(13)

which plays the role of an Israel–Stewart-type relaxation equation for the heat flux [24], where $\Xi \ge 0$ is a transport coefficient. Indeed, if we contract (12) with $n^{\nu }$ and use Equation (13), we find that the fluid produces entropy:

$${\nabla }_{\mu }{s}^{\mu }=-{\displaystyle \frac{\Xi }{n^{\lambda }{w}_{\lambda }}}\left(g{}_{\nu }^{\mu }-{\displaystyle \frac{n^{\mu }{n}_{\nu }}{n^{\alpha }{n}_{\alpha }}}\right)s^{\nu }{w}_{\mu }.$$

(14)

The non-negativity of the entropy production follows from the observation that the term in the round brackets on the right-hand side is the spacelike projector orthogonal to $n^{\mu }$, and $B{s}^{\nu }=(BC-A^2)w^{\nu }+A{n}^{\nu }$, so that

$${\nabla }_{\mu }{s}^{\mu }=-{\displaystyle \frac{\Xi }{n^{\lambda }{w}_{\lambda }}}{\displaystyle \frac{BC-A^2}{B}}\left(g{}_{\nu }^{\mu }-{\displaystyle \frac{n^{\mu }{n}_{\nu }}{n^{\alpha }{n}_{\alpha }}}\right)w^{\nu }{w}_{\mu }\ge 0.$$

(15)

4. Recovering the GENERIC Theory

Let us now show that the multifluid model described above is mathematically equivalent to the GENERIC theory for heat conduction constructed by Öttinger [26], just expressed using different notations.

4.1. Decomposition of Tensors

The GENERIC theory adopts the Eckart frame, meaning that it defines a preferred four-velocity $u^{\mu }$, aligned with the particle current: $n^{\mu }=n{u}^{\mu }$. The temperature and chemical potential are defined through the equations $T=-u^{\mu }{w}_{\mu }$ and $\mu =-u^{\mu }{\xi }_{\mu }$. The vectors $u^{\mu }$ and $w^{\mu }$ are treated as primary vectors, meaning that all other tensors are geometrically decomposed as linear combinations of tensor products of $\{u^{\mu },w^{\mu },g^{\alpha \beta }\}$. For example, $s^{\mu }$ and ${\xi }^{\mu }$ are expressed as

$$\begin{array}{cc}\hfill s^{\mu }=& (s-\sigma T)u^{\mu }+\sigma w^{\mu },\hfill \\ \hfill {\xi }^{\mu }=& {\displaystyle \frac{\epsilon +P-\sigma T^2}{n}}{u}^{\mu }-{\displaystyle \frac{s-\sigma T}{n}}{w}^{\mu }.\hfill \end{array}$$

(16)

Comparing (16) with (11), we obtain the dictionary relations below:

$$s=nA{B}^{-1}+\sigma T\sigma =C-A^2{B}^{-1}\epsilon =n^2{B}^{-1}-P+\sigma T^2.$$

(17)

Since the inverse entrainment matrix is positive definite by stability, we have that $\sigma >0$ and $\epsilon +P>0$. If we substitute (16) into the multifluid formula for the stress–energy tensor as given in (11), we obtain

$$T^{\alpha \beta }=P{g}^{\alpha \beta }+(\epsilon +P-\sigma T^2)u^{\alpha }{u}^{\beta }+\sigma w^{\alpha }{w}^{\beta }.$$

(18)

In accordance with the Eckart frame definition, we have that n and $\epsilon$ are indeed the particle and energy density as measured in the reference frame defined by $u^{\mu }$, namely $n=-n^{\mu }{u}_{\mu }$ and $\epsilon =T^{\alpha \beta }{u}_{\alpha }{u}_{\beta }$.

4.2. First Law of Thermodynamics

In the GENERIC representation, the differential (9) at fixed metric components becomes

$$dP=-\left[(s-\sigma T)u^{\mu }+\sigma w^{\mu }\right]d{w}_{\mu }-n{u}^{\mu }d{\xi }_{\mu }.$$

(19)

We can use the Leibnitz rule to rewrite this differential as follows:

$$dP=sdT+nd\mu -{\displaystyle \frac{\sigma }{2}}d(w^{\mu }{w}_{\mu }+T^2)+(s{w}_{\mu }-\sigma T{w}_{\mu }+n{\xi }_{\mu })d{u}^{\mu }.$$

(20)

The second equation of (16) can be rewritten in the form $s{w}_{\mu }-\sigma T{w}_{\mu }+n{\xi }_{\mu }=(\epsilon +P-\sigma T^2)u_{\mu }$ so that the last term in (20) vanishes. Furthermore, if we contract the second equation of (16) with $u_{\mu }$, we obtain $\epsilon +P=Ts+\mu n$, which is in agreement with Equation (50) of [26]. This allows us to derive the following differential:

$$d\epsilon =Tds+\mu dn+{\displaystyle \frac{\sigma }{2}}d(w^{\mu }{w}_{\mu }+T^2).$$

(21)

Our Equations (16), (18), and (21) are indeed consistent with Equations (39), (45), (54), (55), and (65) of [26] (note that Öttinger [26] splits the energy into rest mass part plus internal contributions, while we are combining them together; thus, the notation “${\rho }_{\mathrm{f}}{c}^2+{\epsilon }_{\mathrm{f}}$” of [26] becomes just $\epsilon$ in our notation); this shows that the constitutive relations of the GENERIC theory can be recovered from the multifluid formalism, by simply expressing all equations in the Eckart frame.

4.3. Recovering the GENERIC Field Equations

To complete our analysis, we only need to verify that Equations (13) and (14) are indeed the dissipative equations of motion of the GENERIC theory. To this end, let us set $\Xi =n/\tau$, where $\tau$ is the heat relaxation time. Then, (13) and (14) become

$$\begin{array}{cc}\hfill & u^{\mu }({\nabla }_{\mu }{w}_{\nu }-{\nabla }_{\nu }{w}_{\mu })=-{\displaystyle \frac{1}{\tau }}(w_{\nu }-T{u}_{\nu }),\hfill \\ \hfill & T{\nabla }_{\mu }{s}^{\mu }={\displaystyle \frac{\sigma }{\tau }}(w^{\mu }{w}_{\mu }+T^2),\hfill \end{array}$$

(22)

which indeed coincide with Equations (73) and (75) of [26]. This completes the correspondence between the GENERIC and the multifluid theory. Note that no approximation was made, meaning that the “dictionary” is exact in the fully non-linear regime. For completeness, we summarize the GENERIC equations in Appendix A.

5. Infinite Heat Conduction Becomes Superfluidity

As an application of the above result, let us show that if $\Xi \to 0$ (which corresponds to sending the heat conductivity to infinity), the GENERIC theory becomes indistinguishable from the two-fluid model for the superfluidity of Son [67] for certain initial conditions. Our starting point is the established fact [64] that a superfluid is a multifluid of Carter, where the equation of motion (13) is replaced by the irrotationality condition ${\nabla }_{[\mu }{\xi }_{\nu ]}=0$; see Appendix B for the proof. Hence, we just need to set $\Xi \equiv 0$ in (13) and (14) and verify that superfluid dynamics are recovered.

5.1. Rigorous Proof

For $\Xi =0$, the GENERIC equations of motion reduce to

$$\begin{array}{cc}\hfill & {\nabla }_{\mu }{n}^{\mu }=0,\hfill \\ \hfill & {\nabla }_{\mu }{s}^{\mu }=0,\hfill \\ \hfill & 2n^{\mu }{\nabla }_{[\mu }{w}_{\nu ]}=0,\hfill \\ \hfill & 2s^{\mu }{\nabla }_{[\mu }{w}_{\nu ]}+2n^{\mu }{\nabla }_{[\mu }{\xi }_{\nu ]}=0.\hfill \end{array}$$

(23)

As we can see, the fluid is reversible. However, it is not exactly superfluid because the third equation differs from the superfluid requirement that ${\nabla }_{[\mu }{\xi }_{\nu ]}$ should vanish. However, let us assume that on an initial Cauchy surface,

$${\nabla }_{[\mu }{w}_{\nu ]}={\nabla }_{[\mu }{\xi }_{\nu ]}=0.$$

(24)

Then, inserting the third equation of (23) into Cartan’s magic formula,

$${\mathcal{L}}_{\mathbf{n}}\left(dw\right)={\iota }_{\mathbf{n}}{d}^{2}w+d\left({\iota }_{\mathbf{n}}dw\right)=0,$$

(25)

we find that ${\nabla }_{[\mu }{w}_{\nu ]}$ vanishes across the whole spacetime. Hence, the fourth equation of (23) becomes $2n^{\mu }{\nabla }_{[\mu }{\xi }_{\nu ]}=0$. Again, using the initial condition (24), Cartan’s magic formula,

$${\mathcal{L}}_{\mathbf{n}}\left(d\xi \right)={\iota }_{\mathbf{n}}{d}^2\xi +d\left({\iota }_{\mathbf{n}}d\xi \right)=0,$$

(26)

guarantees that ${\nabla }_{[\mu }{\xi }_{\nu ]}$ also vanishes everywhere. In conclusion, we have that, under the initial condition (24), GENERIC fluids with infinite heat conductivity are exact solutions of the superfluid equations of motion.

The above proof is a multifluid generalization of the Helmholtz theorem, according to which an inviscid fluid that is initially irrotational remains irrotational at later times. Unfortunately, ${\nabla }_{[\mu }{w}_{\nu ]}$ is not necessarily zero in a superfluid. Hence, not all superfluid configurations are reproduced by the GENERIC theory. However, in most statistical mechanical calculations, one starts from a fluid in homogeneous equilibrium, which is then disturbed by a gravitational perturbation to the metric. Under these conditions, (24) holds, and the heat conductive fluid responds like a superfluid to fully non-linear disturbances.

5.2. Intuitive Explanation

The above result is the non-linear generalization of a recent analysis carried out in [68]. There, it was found that close to homogeneous equilibrium, the infinite conductivity limit of the linearized Israel–Stewart theory for heat conduction [1] is mathematically equivalent to the linearized Landau two-fluid model for superfluidity (see [30], §141). This seems rather counterintuitive, as one would expect that, by increasing the diffusivity of the system, we increase dissipation, which is at odds with the non-dissipative character of a superflow. To understand this phenomenon, let us consider a simplified one-dimensional model.

For incompressible flows, the thermal sector of the linearized Israel–Stewart theory can be approximated by the Cattaneo equation,

$${\displaystyle \frac{{\partial }_t^{2}T}{v^2}}+{\displaystyle \frac{{\partial }_{t}T}{D}}={\partial }_x^{2}T,$$

(27)

where T is the temperature, D is the thermal diffusivity (proportional to the heat conductivity), and v is the speed of propagation of information. Now, if $D\to +\infty$, the second term in (27) vanishes. On the other hand, the speed v cannot diverge together with D because the principle of causality requires that $v^2\in [0,1]$ and causality is necessary for stability [69,70,71]. Therefore, Equation (27) must become a non-dissipative wave equation:

$${\partial }_t^{2}T=v^2{\partial }_x^{2}T.$$

(28)

This tells us that when $D=+\infty$, the system becomes reversible, and heat travels like an undamped wave. As it turns out, Equation (28) is also the propagation equation of the second sound in superfluids [30].

Figure 1 shows the behavior of the dispersion relations of the Cattaneo theory as D increases, keeping v constant. Sending $D\to +\infty$ while holding v constant may seem to be a rather artificial limit, but it is actually the most natural one. In fact, the speed of information is given by the equation $v^2=D/\tau$. Both D and $\tau$ usually scale like $g^{-1}$, where g is the coupling constant determining the strength of the mechanism at the origin of the relaxation (e.g., the particle cross-section in a gas [7,43,72] or the quasi-particle interaction in a quantum liquid [73,74]). Thus, if we send g to zero, D diverges, while v remains constant, being independent of g.

The key to resolving the apparent paradox of large diffusion without dissipation is causality. If we increase the diffusivity, heat spreads faster and faster (the diffusive mode initially moves down in the complex plane). However, energy cannot spread superluminally. For this reason, at some point (when the diffusive mode meets the non-hydrodynamic mode), heat will no longer be able to diffuse in a Gaussian manner, and it will develop a front, which carries energy like a wave (both frequencies acquire a real part). If we further increase the diffusivity, we just cumulate more heat on the front. Eventually, all the energy travels like a wave, and the propagation becomes reversible.

We remark that a similar type of “second sound” has already been observed in solids [77,78,79,80]. Given that most liquids exhibit solid-like behaviors at short distances (see, e.g., [41,81]), it may be possible to observe the transition in Figure 1 experimentally also in fluids. To this end, one needs to make the relaxation time $\tau$ (and therefore D) parametrically large, effectively turning on wave-like dynamics; this can be achieved through confinement [82].

6. Conclusions

We have proved that there is an exact duality between the GENERIC theory for heat conduction [26] and Carter’s two-fluid theory [19]. This mathematical correspondence can be used to derive the constitutive relations of relativistic fluids non-perturbatively, directly within the GENERIC framework [25]. Below, we propose a formal setup that should allow one to compute the GENERIC constitutive relations from microphysics.

Let us consider a fluid in hydrostatic equilibrium, crossed by a time-independent flow of heat in the direction $x^1$. Assuming invariance under translations in the directions $x^2$ and $x^3$, we can solve the first equation of (22), and we find that $w_1=-\tau {\partial }_{1}T$. Furthermore, from Equation (18), we have that the heat flux in the GENERIC theory is given by $q^1=T^{01}=T\sigma w^1$. Considering that $\sigma$ is itself a (even) function of $w^1$, we obtain the following identity:

$${\displaystyle \frac{q^1}{T}}=-\sigma \left(\tau {\partial }_{1}T\right)\tau {\partial }_{1}T.$$

(29)

This can be interpreted as the non-linear generalization of Fourier’s law. If we have a microscopic formula for $q^1$ in this simple setup (derived, e.g., from kinetic theory), we can use it to obtain the constitutive relation $\sigma \left(\tau {\partial }_{1}T\right)$ by direct matching. From this, we can reconstruct the GENERIC equation of state in the non-perturbative regime.

For illustrative purposes, let us consider an analytical (1+1)-dimensional kinetic toy model.

Assume that the heat is transported by quasi-particles with speed 1 and average lifetime $\tau =\mathrm{const}$. Then, we can write the following (approximate) kinetic formula for the heat flux in the origin:

$$q^1\left(0\right)=-{\int }_0^{+\infty }\left[\Sigma \left(x\right)-\Sigma (-x)\right]e^{-x/\tau }dx$$

(30)

This equation tells us that the fluid element at location $\pm x$ is sent towards the origin quasi-particles that move with velocity $\mp 1$, each of which has probability $e^{-x/\tau }$ of reaching the origin.The source term $\Sigma$ needs to be prescribed from microphysics. If we choose, say, $\Sigma =e^{T\left(x\right)}\approx e^{T\left(0\right)+x{\partial }_{1}T\left(0\right)}$, the integral converges for $|\tau {\partial }_{1}T|<1$, and we find

$$T\sigma ={\displaystyle \frac{2\tau e^T}{1-{\left(\tau {\partial }_{1}T\right)}^2}}.$$

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This formula is by no means realistic, but it serves as proof of principle to show that it is possible to compute deviations from Fourier’s law using non-perturbative techniques. Such deviations can then be implemented into Carter’s theory using the duality with the GENERIC theory.

We remark that while the GENERIC theory has the advantage of being more closely related to statistical mechanics, Carter’s framework is a remarkably versatile formalism due to its close connection with exterior calculus [66,83,84,85]. Thus, the present duality may also be useful for those who wish to study the mathematical properties of the GENERIC theory, as we carried out in Section 5.