Fluctuating relativistic dissipative hydrodynamics as a gauge theory

Relativistic hydrodynamics has proven to be an immensely successful phenomenological approach, effective for systems as small as hadronic collisions, and as large as the universe as well. Its theoretical origin is in principle clear. It arises as an effective theory for a system with a “high density of particles” which also thermalizes on a small scale with respect to the microscopic gradients. The deviations from full thermalization could then be incorporated in dissipative coefficients that are associated with gradients of hydrodynamic variables. Nevertheless, a comprehensive account of how hydrodynamics emerges from statistical mechanics is still lacking. The relationship between hydrodynamics as a deterministic theory of conservation laws and statistical mechanics, which explicitly relies on microscopic fluctuations, is in tension. Based on statistical mechanics, we may be able to determine “how far from local equilibrium” the system has to be before hydrodynamics no longer serves as an approximate description. As a consequence of the success of the hydrodynamic description on comparatively small and high gradient systems, such as proton-proton collisions, these issues need to be addressed [1, 2, 3].

Hydrodynamics as an effective theory relies on only conserved currents, whose total flux commutes with the microscopic Hamiltonian [4, 5, 6, 7, 8, 9, 10, 11]. In the absence of chemical potentials, the only conserved current is the energy-momentum tensor, $T^{\mu\nu}$. In spite of this, the assumption of approximate local equilibration still requires the local rest frame (LRF), which itself requires defining the four-velocity fluid (or fluid flow), which are additional details. This is a result of the ergodic assumption that time averages can be approximated by ensemble averages, a fundamental assumption in statistical mechanics, and the fact that time averaging is frame-dependent under special relativity. Considering fluid dynamics as a coarse-grained description, one divides the medium into space-time domains (fluid cells) for which the macroscopic quantities are approximately constant, but their size is much larger than the microscopic physics scale. In the LRF, a thermodynamic limit is a good approximation within the cell if the fluid is non-dissipative. The LRF represents the comoving frame in each cell. Nevertheless, out of equilibrium, the definition of the fluid four-velocity and the LRF are ambiguous. To remove this ambiguity, one traditionally uses the Landau [12] or Eckart [13] definitions for hydro frame selection¹¹1The reader should be aware of the terminology here. Ref. [14] explains hydro frames in detail.. These two options are identical if the fluid is uncharged, which is the assumption we have made throughout this study. First-order hydrodynamics, the so-called Navier-Stokes (NS) hydrodynamics, is known to be unstable for both choices²²2In this work, the term NS is exclusively used for the first-order hydrodynamics in Landau and Eckart frames. [15, 16, 17]. In particular, the Landau frame is defined by

$$u_{\mu}T^{\mu\nu}=eu^{\nu}\phantom{AA},\phantom{AA}\Delta^{\mu}_{\alpha}u_{\beta}T^{\alpha\beta}=0\,,$$

(1)

wherein $e$ is the equilibrium energy density, $\Delta^{\mu\nu}\equiv g^{\mu\nu}-u^{\mu}u^{\nu}$, and $u^{\mu}$ is the fluid four-velocity. This leads to the transversality condition

$$u_{\mu}\Pi^{\mu\nu}=0$$

(2)

where $\Pi^{\mu\nu}$ contains all out of equilibrium corrections to the energy-momentum tensor. With first-order hydrodynamics, one introduces terms of first-order in derivatives of hydrodynamic variables, coupled to new transport coefficients. The resulting energy-momentum tensor can be written as

$$T_{\mu\nu}=(e+p)u_{\mu}u_{\nu}+pg_{\mu\nu}+q_{\mu}u_{\nu}+q_{\nu}u_{\mu}+\left.\Pi_{\mu\nu}\right|_{\propto\partial u}\,,$$

(3)

wherein $q_{\mu}$ is a vector transverse to $u_{\mu}$. Typically, one uses the equations of motion from the zeroth-order to eliminate the so-called off-shell terms, i.e. terms that vanish when the equations of motion are satisfied. Based on the Kubo formulae, it is possible to derive the transport coefficients from microscopic theory [18]. At tree-level, these formulae coincide with estimates from the Boltzmann equation.

As of recently, Israel-Stewart (IS) hydrodynamics has been the main approach to link hydrodynamics with far-from-equilibrium systems and remedy the instability of NS hydrodynamics [19]. In this approach, new dynamical variables are introduced into the nonequilibrium part of the energy-momentum tensor, $\Pi^{\mu\nu}$, which relax to their NS counterparts on a time scale quantified by the theory’s relaxation time. Schematically

$$\tau_{X}u^{\mu}\partial_{\mu}X+X=X_{\rm NS}\,,$$

(4)

in which $X$ is a hydrodynamic variable expressed in terms of an arbitrary rank tensor. More complicated, oscillatory behavior is also possible if one adds second-order derivatives in time [20]. These theories are usually treated as ”top-down effective theories”, with the ”UV completion” being either a weakly coupled Boltzmann equation or a strongly coupled theory with a gauge dual. However, we know from condensed matter physics [21] that hydrodynamics emerges from a much wider and richer range of systems, notably systems which are strongly correlated at a short scale and more advective-diffusive at a longer one.

Recently, a new class of theories was developed, the so-called Bemfica-Disconzi-Noronha-Kovtun (BDNK) theory, where $\Pi_{\mu\nu}$ is entirely determined by gradients of the hydrodynamic variables [22, 23, 24, 25, 26, 27], based on earlier proposals [28, 29, 30] which reduce the degrees of freedom at the expense of local isotropy. Numerical simulations [31] have confirmed that when the Knudsen number is small, and only in that limit, do the two theories coincide, and this has been interpreted as the statement that while BDNK-type theories might be numerically easier to simulate they would not provide an explanation to the “unreasonable effectiveness of hydrodynamics“.

There are some basic differences between the BDNK first-order theory and traditional NS hydrodynamics. First, in the BDNK theory, the eigenvalues of neither the energy-momentum tensor nor the current correspond to physical quantities since $u_{\mu}$ is a somewhat arbitrarily chosen vector field. Second, the off-shell terms in the gradient expansion are kept. The price is an uncertain relation between microscopic thermodynamics, entropy, and hydrodynamic evolution since it is unclear what is the right ensemble to sample to get the equation of state and transport coefficients. In addition, the entropy production density $\partial_{\mu}S^{\mu}$ is not always positive (sometimes $\partial_{\mu}S^{\mu}<0$ for large gradients [32, 33, 34]). This last point could be rationalized by assuming that for large gradients the entropy gradient is not the one obtained from a gradient expansion. However, given that fluctuation and dissipation are correlated, transient negative entropy intervals are not necessarily unphysical by themselves, they just reflect stages in evolution where thermodynamic fluctuations are non-negligible.

To properly understand these ambiguities, one must realize that the fluid four-velocity $u_{\mu}$, energy density $e$, fluid pressure $p$, the heat current $q_{\mu}$, and shear tensor $\pi_{\mu\nu}$ may be considered unobservable figments of the theorists’ imagination, since what is actually observed is the entire $T_{\mu\nu}$. The effect hydrodynamic fluctuations have on these dynamics, particularly on $\Pi_{\mu\nu}$, heighten this ambiguity [3]. Ultimately, experimental observables (such as elliptic flow and higher harmonics) reduce to correlations $\left\langle T_{\mu_{1}\nu_{1}}(x_{1})...T_{\mu_{n}\nu_{n}}(x_{n})\right\rangle$. Part of the approximation of local equilibration is that we cannot tell which “microscopic particle” (molecule, quark, gluon) contributes to “equilibrium” energy density, pressure, $u_{\mu}$ and which is “not thermalized” and contributes to $\Pi_{\mu\nu}$.

The choice of frame is, therefore, more similar to “gauge fixing” than to a real physical distinction. As a related point, Fig. 1 illustrates that the relative contribution of thermal fluctuations and thermodynamic evolution to a given observable is experimentally undetectable. The different ways the two contributions contribute give a physical justification for this “gauge symmetry”. The choice between a Landau, Eckart, or a more anisotropic frame such as BDNK represent a choice of the role of deterministic dissipative evolution versus thermal fluctuations in how a coarse-grained final state evolved from a similarly coarse-grained initial state. Microscopically this ”choice” can be associated with Bayesian inferencing weighted by Gibbsian entropy [35] over an ensemble of hydrodynamic frames. In terms of macroscopic variables, a range of such frame choices determined by each choice’s unique local thermal fluctuations and evolution of conserved quantities should be physically equivalent.

This could, in principle, provide an explanation for the unreasonable effectiveness of hydrodynamics [1], provided the effectiveness of hydrodynamics is not anymore judged by the Knudsen number and averaged gradients (in an ensemble average that breaks down for few particles) but rather by the local applicability of the fluctuation-dissipation theorem, i.e. the idea that dynamics is determined by the number of locally available microstates. In the limit where interactions are “strong“ and microscopic particles are strongly correlated (an everyday example is the “Brazil nut effect” [36]), it is not guaranteed that a system with more degrees of freedom achieves a more perfect local thermal equilibrium and close-to-ideal fluid behavior. As the number of degrees of freedom decrease, fluctuations increase but so do redundant descriptions which are compatible with local equilibrium but have different Boltzmannian (although not Gibbsian) entropies. Thus, the chance of finding a description that happens to look like ideal nearly hydrodynamics is greater.

The authors are aware of one other hydrodynamic formalism which produces causal and stable evolution, namely the divergence type hydrodynamic equations [37, 38, 39]. In this work we will not make many connections with this formalism, other than a possible relation in Sec. V. We leave for future work to include the effects of hydrodynamic frame and their associated redundancies into a fluctuating description of divergence type theories [40, 41].

The rest of the paper will outline how the frame choice ambiguity outlined above can be seen as a freedom of reparametrization similar to gauge symmetry. First, in the next section II, we review Zubarev hydrodynamics and the abiguity of the choice of foliation, reflected in the ambiguity of the flow vector $\beta_{\mu}=u_{\mu}/T$. In Sec. III we define the reparametrization linking microscopic fluctuations and dissipative evolution and discuss the details of how it is implemented. We then devote three subsections to it’s physical and microscopic meaning, respectively in Zubarev hydrodynamics (subsection III.1), transport theory (subsection III.2 where the connection to the Gibbsian vs. Boltzmannian entroy current is elucidated) and effective theory (subsection III.3), concluding with considerations on how the reparametrization symmetry affects the causality of the theory (subsection III.4)

Then in Sec. IV we investigate the particular case of a bulk viscous fluids. In subsections IV.1 and IV.2 we show how, in this case, one may relate BDNK like-theories with Israel-Stewart ones by relabeling heat currents as fluctuation-triggered sound waves (and vice-versa), both considered as “microscopic” under a Gibbsian picture. Section IV.3 shows how these relabelings can be seen as a particular case of the reparametrizations described in section III. Finally, in Sec. V, we explore how the symmetries we consider appear as quasi-conserved currents in the effective action approach to hydrodynamics pioneered in [42]

Throughout the paper we adopt the metric signature $(+,-,-,-)$ and work in natural units $c=k_{B}=\hbar=1$.

Hydrodynamics is usually thought of as an effective theory. This begs the question of what the “fundamental theory” is. In the previous section, we cited Gauge/gravity duality and the Boltzmann equation. The latter is obviously more directly connected to physically realistic systems, and indeed most derivations of hydrodynamics employ it. In some approaches, this has been done in the “Wilsonian” renormalization group sense [43, 44]. In this respect, anisotropic frames until now were defined in terms of relaxation time matching conditions of the Boltzmann collision term [45, 46] (an exact renormalization group treatment requires a frame isotropic w.r.t. entropy [44]). Such derivations however are incomplete since the Boltzmann equation is itself incomplete as a theory. There are fluids, such as water, where hydrodynamics itself provides a better description than the Boltzmann equation. Moreover the ”best” result that shows that the Boltzmann equation is a good approximation for any system (in other words that microscopic reversible correlations are irrelevant) is Lanford’s theorem [47], a statement of the time-scale of applicability that notably fails to give a long-time limit for hightly correlated systems.

A different way for hydrodynamics, in particular it’s generic frame form, to emerge is therefore desireable. The most universal ”UV theory” might be the fundamental principle of statistical mechanics, the equal distribution of microstates. As this section and the next one make clear, some care is needed to define the theory this way if the system is not in perfect local equilibrium, i.e. if microstates are not quite equally probable. We shall argue in the next section that the fluctuation-dissipation theorem, promoted to a ”symmetry” can be used in this regime, and anisotropic hydrodynamics emerges as a byproduct of this symmetry. Here, we shall give a brief introduction to Zubarev hydrodynamics, which has the advantage of being defined ab initio via equilibrium and microstate equality. Fundamentally, equilibrium is defined in terms of time-symmetric microscopic states, via the KMS condition [48], occupied uniformly under the constraint of conservation laws.

In the coarse-graining of the fluid one divides the medium into spacetime domains, the so-called fluid cells, for which the macroscopic quantities are approximately constant, but their size is much larger than the microscopic physics scale. If the fluid is non-dissipative, then for the comoving observer, the thermodynamic limit³³3Intended as extensivity, rather than as limit where fluctuations are negligible; this paper applies in the situation where they are not negligible with respect to the mean free path is a good approximation within the cell. Hence such an observer can in principle derive the macroscopic observables from an underlying microscopic theory. Such an underlying theory in the most fundamental level may be encoded in a Lagrangian and therefore the usual methods of thermal field theory can be employed. For example (we neglect the chemical potential throughout this work for simplicity, and therefore Landau and Eckart frames are identical):

$$e=-\frac{\delta}{\delta\beta}\ln\mathcal{Z}\phantom{AA},\phantom{AA}p=\beta^{-1}\ln\mathcal{Z}\,.$$

(5)

Here $\mathcal{Z}$ is the partition function, and $\ln\mathcal{Z}$ is obtainable from a functional integral of the microscopic Lagrangian⁴⁴4The equilibrium part being defined using the Matsubara time prescription, which stipulates that time is periodic [49, 50]

$$\mathcal{Z}=\int\mathcal{D}\phi\exp[S[\phi]]\,,$$

(6)

wherein $S[\phi]$ is the Euclidean periodic action. The Landau frame is special in the sense that Eq. (1) continues to hold for the dissipative fluid. In the grand canonical ensemble, neglecting quantum corrections and correlations as well as chemical potentials for simplicity, we have

$$\mathcal{Z}=\sum_{i=0}^{\infty}\mathcal{Z}_{N}\phantom{AA},\phantom{AA}\mathcal{Z}_{N}=\frac{1}{N!}\int\prod_{j=1}^{N}d^{3}p_{j}\,d^{3}x_{j}\exp\left[-\frac{E_{i}(p_{1},...,p_{N})}{T}\right]$$

(7)

The Gibbs-Duhem relation

$$s\equiv\frac{dp}{dT}=\frac{p+e}{T}\phantom{AA},\phantom{AA}T=\beta^{-1}$$

(8)

follows automatically from Eq. (5) and guarantees that the equilibrium entropy current is conserved, $\partial_{\mu}\left(su^{\mu}\right)=0$. The above definitions are quite basic, in that they directly follow from the basic assumptions of statistical mechanics. In that respect, in equilibrium LRF is “special”. However, this frame can only be unambiguously known if “neighboring cells” used to coarse-grain the fluid (separated by a sound-wave dissipation length) are also each in the thermodynamic limit. This is necessary so that thermal fluctuations factor out, otherwise neighboring cells would drive the cell of interest out of equilibrium which is a manifestation of the fluctuation-dissipation theorem. But this assumption is stronger than the assumption that the Knudsen number ⁵⁵5Knudsen number is the ratio of particles mean free path to the typical length scale for which the hydrodynamic variables change. is small, and can not hold for small systems even if gradients/Knudsen numbers are small. On the other hand, those fluctuation-dissipation relations (on which linear response theory is based) are independent of ensemble choice [51, 52, 53] and so local equilibration could occur without the necessity of assuming that every cell has to be large enough to approach the thermodynamic limit⁶⁶6Generally [51], Eq. (7) can be moved to a microcanonical ensemble via $$\ \Omega(E)=\int dT\mathcal{Z}(T)\exp\left[iTE\right]$$ equations 9 will be updated straight-forwardly via linear dispersion relations [53]. Temperature fluctuations $\sim T^{2}/c_{V}$ while energy fluctuations $\sim c_{V}/T^{2}$, where $c_{V}$ is the second derivative of Eq. 7 w.r.t. T [50].. While, since cell coarse-graining is arbitrary, one expects the grand canonical ensemble to apply for fluids, changing between ensembles is in principle straight-forward [51, 52], one can also appropriately convert Eq. (7) if necessary.

Given a field configuration close to thermal equilibrium and a contour respecting energy-positivity and the KMS condition, transport coefficients can be obtained from infrared limits, i.e. $k\to 0$, of the Eq. (10) of the energy-momentum tensor [48, 54, 18, 55, 56], using the fluctuation-dissipation relation

$$\eta\sim\lim_{k\rightarrow 0}\mathrm{Im}\frac{1}{k}\tilde{G}(k)\phantom{AA},\phantom{AA}\tau_{\pi}=\lim_{k\rightarrow 0}\frac{k}{2\mathrm{Im}[G(k)]}\left[\frac{1}{k^{2}}-\frac{1}{w(k)^{2}}\right]\mathrm{Re}[G^{\prime}(k)]\,,$$

(9)

wherein

$$\tilde{G}(k)=\int\left.\theta\left(\mathrm{Im}\left(x^{0}-x^{0^{\prime}}\right)\right)e^{ik(x-x^{\prime})}\frac{1}{g}\frac{\delta^{2}}{\delta g_{ij}^{+}(x)\delta g_{ij}^{-}(x^{\prime})}\ln\mathcal{Z}\,.\color[rgb]{0,0,0}\right|_{g_{\mu\nu}=\eta^{\pm}_{\mu\nu^{\prime}}}$$

(10)

and

$$\ \eta^{\pm}_{\mu\nu}=diag\left(\frac{1}{T}\exp\left[\frac{i2\pi\theta}{T}\right]\pm i\epsilon,-1,-1,-1\right)$$

is the Minkowski metric with a periodic boundary condition enforcing Matsubara time. Note that all microscopic interactions in the above are in principle calculable via $E(p_{i},x_{i})$, the Hamiltonian.

However, we note a tension between the development of “macroscopic causal” hydrodynamics described in the previous section, and the considerations of statistical mechanics done here. The fact that in Eq. (3) the “LRF” parameters depend on $u_{\mu}$ means that the very definition of the equation of state and transport coefficients might be not-trivial, although it might be irrelevant for microscopic evolutions in certain limits [15, 57]. Even in the Landau frame, where we expect Eq. (9) to apply straight-forwardly, the positivity of the entropy production relies on the assumption that the ratio $\tau_{\pi}/(T\eta)$ is the second-order expansion coefficient of the entropy density in terms of $\Pi_{\mu\nu}$, and hence is a merely thermodynamic (not kinetic) quantity [15, 57], arising from the pole structure of thermodynamic correlators [58] . However, as [18, 59] and earlier works show, the relaxation time can be seen as a transport coefficient, given by a Kubo formula. The tension between these pictures reflects the fact that in an interacting theory “thermodynamic” and “transport” quantities cannot be straightforwardly unentangled. This is the basic reason why transport perturbative calculations exhibit divergences even in weakly coupled theory [60] and also reflects the fact that “local thermodynamic limit” and “small Knudsen number” refer to different limits and thus need to be handled consistently in a hydrodynamic treatment. These ambiguities are swept under the carpet in the Knudsen number expansion because it is implicitly assumed the gradient is large enough w.r.t. the scale setting the thermodynamic limit that $k\to 0$ in Kubo formulae can consistently render fluctuations irrelevant. This is not true in systems of $\mathcal{O}\left(50\right)$ particles.

As a related issue, strict positivity of the entropy production only holds “on average” over “large distances” [3]. Beyond this, one has a fluctuation-dissipation theorem [48, 54], which is more directly related to unitarity and time-reversal invariance of the underlying microscopic theory [54]. Entropy increase and its maximization in equilibrium should emerge in the “far-infrared hydrodynamic scale” w.r.t. microscopic fluctuations. However, turbulence and anomalous dissipation [61] make it likely that microscopic fluctuations back-reacting on hydrodynamic modes will modify infrared transport coefficients (as corrections from interacting sound waves generally do [62, 14, 63]), and the fact that this must be done in a way that maintains causality and stability for all possible effective theories suggest microscopic-macroscopic correlations need to be systematically understood, and also that such correlations mean equivalent descriptions exist where “sound wave backreaction” are replaced by “thermal fluctuations” (and vice-versa).

Because of the fact that any interaction potential will make Gibbsian and Boltzmannian entropy different [35], the above ambiguity will make the process of equilibration ambiguous, because one could interpret inhomogeneities as correlations present in the partition function or collective excitations that will dissipate and be reseeded as the system evolves. Deterministic hydrodynamics however neglects this, since one usually assumes Boltzmannian entropy current to be the physically relevant one.

Zubarev hydrodynamics makes some of the issues above clearer: if the system is approximately in local equilibrium, one can [3, 64] impose the KMS condition and assume quantum interference is “irrelevant”,i.e. it decoheres much faster than the macroscopic evolution. The first assumption means that the partition function depends not on the microscopic lagrangian but on the energy-momentum tensor $T_{\mu\nu}$ contracted with a field of Lagrange multiplies $\beta^{\mu}=u^{\mu}/T$ and a spacetime foliation d$\Sigma_{\mu}$. The second assumption means that the foliation $d\Sigma_{\mu}$ can be trivially analytically continued. The partition function is then

$$\mathcal{Z}_{E}(\Sigma,\beta_{\mu})=\int\mathcal{D}\phi\exp\left[-\int_{\Sigma(\tau)}d\Sigma_{\mu}\beta_{\nu}\hat{T}^{\mu\nu}\right]$$

(11)

Since the density matrix is determined by the partition function and boundary conditions [65], it is easy to see that, at least if one neglects boundary effects, Eq. (11) produces a density matrix stationary in the frame co-moving with $\beta_{\mu}$ provided the system was in perfect equilibrium at the first foliation. This requirement of course can not be generally fulfilled. However, if this requirement is fulfilled approximatly using Stoke’s theorem [64], one can expand the density matrix to the first order around the local equilibrium [64]

$$\mathcal{Z}_{E}(\Sigma,\beta_{\mu})\simeq\int\mathcal{D}\phi\exp\left[-\int_{\Sigma(\tau+d\tau)}d\Sigma_{\mu}\;T^{\mu\nu}\beta_{\nu}+\int_{\partial\Sigma}d\Omega\;T^{\mu\nu}\nabla_{\mu}\beta_{\nu}\right]$$

(12)

where $\partial\Sigma$ is a small rectangle delimiting the foliation $\Sigma(\tau)$ and $\Sigma(\tau+d\tau)$ and $d\Omega$ a surface element of that rectangle. One can see by counting gradients that dissipative dynamics comes from the second term only [64]

Since the partition function and the density matrix contain the same information[65], one would expect to recover the same information of Eq. Eq. (9) using Zubarev techniques. Indeed, eq Eq. (12) was used in  [64] to expand around equilibrium for any operator

$$\langle\widehat{O}(x)\rangle-\langle\widehat{O}(x)\rangle_{\rm LE}\simeq iT\int_{t_{0}}^{t}\!\!\!{\rm d}^{4}x^{\prime}\int_{t_{0}}^{t^{\prime}}\!\!\!{\rm d}\theta\;\left(\langle[\widehat{O}(x),{\widehat{T}}^{\mu\nu}(\theta,{\bf x}^{\prime})]\rangle_{\beta(x)}\partial_{\mu}\beta_{\nu}(x^{\prime})\right)$$

(13)

where $\theta$ refers to the standard imaginary Matsubara time. Using these equations, with $\hat{O}\equiv\hat{T}_{\mu\nu}$, the (non-causal) NS equations can be derived from a gradient expansion, as can the Kubo formulae Eq. (9) for $\eta$ and $\zeta$, as can be seen in [64] which coincide with the metric formulas used in [56]. Recently this approach was extended to second order coefficients [59].

We note that the above formulae, Eq. (11) and Eq. (13) are valid only at a scale where thermal fluctuations can be discounted. One can see this via the gravitational Ward identiy, [66, 18, 67]

$$\partial^{\alpha}\left\{\left\langle\left[\hat{T}_{\mu\nu}(x),\hat{T}_{\alpha\beta}(x^{\prime})\right]\right\rangle-\delta(x-x^{\prime})\left(g_{\beta\mu}\left\langle\hat{T}_{\alpha\nu}(x^{\prime})\right\rangle+g_{\beta\nu}\left\langle\hat{T}_{\alpha\mu}(x^{\prime})\right\rangle-g_{\beta\alpha}\left\langle\hat{T}_{\mu\nu}(x^{\prime})\right\rangle\right)\right\}=0\,,$$

(14)

This identity holds for the “UV” theory (when all correlations are included) at all scales [67] as a consequence of local Lorentz invariance. Provided locally a hydrostatic limit holds and in the infrared limit, the “contact terms” $\sim\left\langle T_{\mu\nu}\right\rangle$ disappear, and the clean separation in Eq. (12) and Eq. (13) can be recovered. When the mean free path and the thermal fluctuation scale are comparable and the background is far from hydrostatic, however, they are present and the two terms of Eq. (12) do not separate, and a clear distinction between “dissipative” and “non-dissipative” terms becomes impossible (physically, thermal fluctuations mix dissipative and non-dissipative contributions).

We can now examine the entropy current issue discussed in the previous section in a bit more detail. As long as the Gibbs stability criterion is respected [68], this non-positivity can be understood by either evoking thermodynamic fluctuations [3] or by altering the form of $\Pi_{\alpha\beta}$ [32]. From the point of view of Zubarev hydrodynamics [64] this can be understood by examining the entropy current’s dependence on the chosen foliation

$$\frac{d}{d\tau}\left(s^{\mu}d\Sigma_{\mu}\right)=\frac{1}{s}s^{\mu}d\Sigma_{\mu}\Pi_{\alpha\gamma}\partial^{\alpha}\beta^{\gamma}\underbrace{\geq 0}_{\rm if\phantom{A}2nd\phantom{A}law}\,,$$

(15)

where $d\Sigma_{\mu}$ is a small 3-volume element (with unit normal $n_{\mu}$), $\tau$ is the proper time w.r.t. that foliation and $s^{\mu}$ the entropy current.

The non-positivity of the entropy production can be readily understood: if one integrated over $d\Sigma_{\mu}$ (as in [64]), entropy would certainly increase, modulo fluctuations going to zero in the infinite volume. But, if $n_{\mu}$ has nothing to do with the entropy current ($u_{\mu}$ in the Landau frame), the foliation is sampling “many cells” rather than tracking the evolution of a single cell. The fluctuation-dissipation effective field theory picture suggested by Fig. 1 suggests these different pictures are linked, in that a variation of $n_{\mu}$ w.r.t. the Landau frame $u_{\mu}$ will lead to decreases in entropy given by a fluctuation-dissipation relation [3]. Thus, the expansion in [23, 24, 25] of Eq. (3) where $q_{\mu}$, $e$, and $p$ in gradients of an arbitrary flow field $u_{\mu}$ under the constraints of the second law is equivalent to a choice of $d\Sigma_{\mu}$ of [64].

Indeed, in Zubarev hydrodynamics $d\Sigma_{\mu}$ is not necessarily parallel to $u_{\mu}$ ⁷⁷7In fact, if $u_{\mu}$ is vorticose it cannot be globally parallel to $u_{\mu}$. Moving away from a parallel frame is equivalent to introducing anisotropy, and to an extent (as long as one loses all touch with equilibrium) it is a theoretical choice, equivalent to redefining $u_{\mu}$ and $\Pi_{\mu\nu}$. In the next section, we will show how to understand the different reparametrizations in terms of symmetries.

As we saw hydrodynamics is usually thought of as an effective theory obtained by coarse-graining a microscopic theory in configurations close to local equilibrium. Thus the usual justification for for choosing different matching conditions [25] is one of “field redefinitions” [45], common to zero temperature effective field theories. This justification must however be be taken with some care as it clashes with fundamental principles of statistical mechanics, such as ergodicity. To clarify the role of redefinitions, thereforre, one must remember that Wilsonian renormalization has a probabilistic interpretation related to the central limit theorem [69]. In this sense, undetectable correlations and fluctuations at or beyond the cut-off scale in effective field theories are “washed out” in the same way that higher cumulants are irrelevant when the sum of many uncorrelated variables is considered. What has recently become appreciated is that for effective theories with UV completion, field redefinitions bring non-trivial correlations sensitive to UV completion (see for instance [70]). In the language of [69], transforming the variables before summing them over can affect how close is the central limit reached.

At finite temperature, for fluids with finite viscosity, these microscopic correlations are complicated by the presence of an entropy current which is approximately, but not quite, locally maximized and does not qualify as a conserved current. [25, 45] formulate the theory in terms of a gradient expansion of conserved quantities, with entropy used as a stability check. This is similar, but not quite the same, as building an effective theory around fluctuation-dissipation from a local equilibrium state, as the local decreases of entropy pointed out in [32] show. As we argue in this work, in fact, the latter approach can also provide a justification for anisotropic hydrodynamics, and in addition clarify the role of microscopic correlations in field redefinitions and a justification of the applicability of hydrodynamics generally different from the smallness of the Knudsen number.

A fluid is defined by its symmetries [71, 7, 4, 5, 8], which in the ideal case are homogeneity, isotropy (as passive transformations: they reparameterize the solution), and volume-dependent diffeomorphisms (as active ones, they turn the solution into another one). Non-ideal fluids break some of these symmetries, but a reparametrization symmetry, evident in the freedom of changing between the BDNK and IS approach, is added. As we shall argue in this section, physically, this symmetry is simply the fact that a correlation in energy-momentum measured at a distance could be the result of a microscopic thermal fluctuation or a macroscopic collective excitation. Measuring this correlation individually gives us no way of knowing its source, and hence many different dynamical configurations could have given rise to it (Fig. 1). Within the Zubarev approach, this symmetry reflects the freedom to choose a foliation. One can imagine a “more time-like foliation” would make deterministic evolution more prominent, and a “more space-like foliation” would enhance thermal fluctuations. But since foliation is a reparametrization symmetry, actual physical observables would not be touched.

Here we try to understand this symmetry from a Lorentz group theory point of view. In general, for the energy-momentum tensor to be decomposed as Eq. (3), one must set up a flow $dM_{\mu\nu}$ (which is just a Lorentz boost)⁸⁸8Note that this discussion changes drastically in the non-relativistic limit, since the group of symmetries there is not a subgroup of the Lorentz group. See last paragraph of section IV.3

$$\Pi_{\mu\nu}\rightarrow\Pi_{\alpha\gamma}\left(g^{\alpha}_{\mu}g^{\gamma}_{\nu}-g^{\alpha}_{\mu}dM^{\gamma}_{\nu}-g^{\gamma}_{\nu}dM^{\alpha}_{\mu}\right)\phantom{AA},\phantom{AA}u_{\mu}\rightarrow u_{\alpha}\left(g^{\alpha}_{\mu}-dM^{\alpha}_{\mu}\right)$$

(16)

as well as the equation of state Eq. (6)

$$\ln\mathcal{Z}\rightarrow\ln\mathcal{Z}+d\ln\mathcal{Z}$$

(17)

$$\ d\ln\mathcal{Z}=\sum_{N=0}^{\infty}\int\prod_{j=1}^{N}d^{4}p_{j}\delta\left(E_{N}(p_{1},...p_{j})-\sum_{j}p_{j}^{0}\right)\sqrt{\mathrm{det}\left[dM\right]}\exp\left(-\frac{dM_{0\mu}p^{\mu}}{T}\right)$$

Note that it is only the $d^{3}p$ integral which is Lorentz transformed at the microscopic level. Instead, the coarse-grained volume interval is just boosted. This is because the volume element Lorentz transformation is taken care of by Eq. (16). Note that this is consistent with the separate (but related) interpretation of $M_{\mu\nu}$ as inducing a thermodynamic fluctuation w.r.t. a flow frame defined by the Landau condition Eq. (1). In terms of the ergodic hypothesis, mentioned earlier, these transformations amount to ”biasing” the local comoving phase space sampling together with a deformation of the local time, to model a local fluctuation in the equilibrium frame. For ideal hydro with negligible fluctuations (large heat capacity), this separation would be impossible since equilibrium phase space is a Lorentz scalar and $\Pi_{\mu\nu}=0$ in Eq. (16).

Eq. (17) is unwieldy (and the factoring out of $dM_{\mu\nu}$ is even more unwieldy, which is why we left the equation in this form) but just reflects equilibrium statistical mechanics in a transformed frame. It is necessary because this transformation is just a reparameterization, leaving the total $\left\langle T_{\mu\nu}\right\rangle$ invariant. Equations (17) and (16) contribute to this, forming a constraint

$$dM_{\mu\nu}(x)\frac{\delta\ln\mathcal{Z}_{E}[\beta_{\mu}]}{dg^{\alpha\mu}(x)}=0$$

(18)

where $\ln\mathcal{Z}_{E}$ is given by Eq. (11). This constraint is nothing but a non-linear generalization of the Gibbs-Duhem relation [72, 19, 68]

$$\delta s^{\mu}=-\beta\mathcal{K}_{\nu}\delta T_{0}^{\mu\nu}=\beta\mathcal{K}_{\nu}\delta\Pi^{\mu\nu}\phantom{AA},\phantom{AA}\mathcal{K}_{\nu}M^{\mu\nu}\propto\mathcal{K}^{\mu}\,.$$

(19)

where $\mathcal{K}_{\mu}$ is a killing vector of the co-moving frame ($\neq u_{\mu}$ away from ideal hydrodynamics). Since transformations defined by $M_{\mu\nu}$ correspond to different foliations they can in principle change entropy and the equilibrium part of the energy-momentum tensor $T_{0}^{\mu\nu}$, but the relation between the two must be invariant.

Because the Boltzmann entropy need not be invariant under parametrizations, the Gibbs stability criterion [72] ($\delta^{1}(s_{\mu}+\delta\beta^{\nu}T_{\mu\nu}=0,\delta^{2}(...)\leq 0$) becomes probabilistic (configurations that recpect it are likely to evolve hydrodynamically, without fluctuations affecting it, while configurations that violate it are likely to be fluctuations-dominated). Thus, as long as such transformations maintain the reference frame close to a comoving equilibrium frame, they are unlikely to violate the Gibbs criterion, with probabilities $P(....)$ of entropy decreasing weighted as

$$\frac{P\left(T_{0}^{\mu\nu}\rightarrow T_{0}^{\mu\nu}+\delta T_{0}^{\mu\nu}\right)}{P\left(T_{0}^{\mu\nu}+\delta T_{0}^{\mu\nu}\rightarrow T_{0}^{\mu\nu}\right)}\sim\exp\left[\mathcal{K}_{\mu}\delta s^{\mu}\right]$$

(20)

in analogy with [3]. Equation Eq. (19) together with the Boltzmann interpretation of entropy implement this condition, which can be taken as a good definition of “close to local equilibrium”.

Since $\ln\mathcal{Z}_{E}$ depends on $\beta_{\mu}$ only, it is clear that these transformations are defined in the comoving frame, so one can automatically satisfy Eq. (18) up to an overall constant by having $dM_{\mu\nu}$ of the form

$$dM_{\mu\nu}=\left(\Lambda\right)^{-1}_{\alpha\mu}\Lambda_{\beta\nu}dU^{\alpha\beta}\,,$$

(21)

where the “kernel” part $dU^{\alpha\beta}$ performs the infinitesimal replacement of Eq. (16). Since, as argued earlier, possible changes can occur in any component of $\Pi_{\mu\nu}$ generators involving all components will be necessary:

$$dU^{\mu\nu}=\eta^{\mu\nu}d\mathcal{A}+\sum_{I=1,3}\left(d\alpha_{I}\hat{J}^{\mu\nu}_{I}+d\beta_{I}\hat{K}^{\mu\nu}_{I}\right)$$

(22)

The first term is the bulk channel, the rest are the shear ones, namely

$$K_{1}=\left(\begin{array}[]{cccc}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array}\right)\phantom{AA},\phantom{AA}K_{2}=\left(\begin{array}[]{cccc}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{array}\right)\phantom{AA},\phantom{AA}K_{3}=\left(\begin{array}[]{cccc}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{array}\right)$$

(23)

which move components from $\Pi_{\mu\nu}$ to $q_{\mu}$ (see Eq. (3)) as well as $J_{1,2,3}$

$$J_{1}=\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&0\\ \end{array}\right)\phantom{AA},\phantom{AA}J_{2}=\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&1&0&0\\ \end{array}\right)\phantom{AA},\phantom{AA}J_{3}=\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&1&0\\ \end{array}\right)$$

(24)

Note that these transformations generally break the transversality condition Eq. (2). Note also that these generators are not arbitrary, because not all combinations solve Eq. (16), Eq. (17) and Eq. (18). Because of the non-linearity of the $\Lambda\Lambda^{-1}$ step in Eq. (21), mapping out allowed valueds of the coefficients is generally not possible, similarly to a non-Abelian gauge constraint equation.

For any starting $\Pi_{\mu\nu}$ equation Eq. (25) can be integrated via a flow parameter $s$

$$\Pi_{\mu\nu}^{\prime}=\int_{s}\left[\eta^{\mu\nu}\frac{d\mathcal{A}}{ds}+\sum_{I=1,3}\left(\frac{d\alpha_{I}}{ds}\hat{J}^{\mu\nu}_{I}+\frac{d\beta_{I}}{ds}\hat{K}^{\mu\nu}_{I}\right)\right]\Pi_{\mu\nu}ds$$

(25)

A fixed point in this evolution, where $\Pi_{\mu\nu}$ is a function of equilibrium quantities and their gradients indicates the transformation of the initial conditions into a BDNK-like “Gauge”. By symmetry this will force $\Pi_{\mu\nu}$ to be a function of the gradient and an an integration constant vector (identified with the heat flow $q_{\mu}$), recovering Eq. (3)

$$\ \Pi_{\mu\nu}\rightarrow q_{\mu}u_{\nu}+q_{\nu}u_{\mu}+f(u,\partial u)$$

A comparison with Eq. (15) as well as Eq. (19) shows that to first order the entropy production at rest with the foliation, $\sim\Pi^{\mu\nu}\partial_{\mu}u_{\nu}$ is zero. Changing $\Pi_{\mu\nu}$ also changes $u_{\mu}$ in a way that that balances out the total energy momentum and the entropy, the latter because the energy taken out of $\Pi_{\mu\nu}$ goes into a ”sound wave created by the thermal fluctuation”. Because of the overall invariance of $T_{\mu\nu}$ (Eq. (18)) these necessarily balance. However, beyond the linearized limit, sound waves will interact, changing that entropy. This change reflects the fact that outside of the ideal fluid limit Gibbsian and Boltzmannian entropy (the latter identified via Eq. (15)) are not the same [35]. Gibbsian entropy, defined to also include sound-wave fluctuations and interactions lost in the coarse-graining, is defined as any entropy missing from the Boltzmannian description. The next two sections give possible ways to understand this entropy quantitatively, in terms of both microscopic and macroscopic perturbations