Do nuclear collisions create a locally equilibrated quark–gluon plasma?
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Abstract
Experimental results on azimuthal correlations in high energy nuclear collisions (nucleus–nucleus, proton–nucleus, and proton–proton) seem to be well described by viscous hydrodynamics. It is often argued that this agreement implies either local thermal equilibrium or at least local isotropy. In this note, I present arguments why this is not the case. Neither local nearequilibrium nor nearisotropy are required in order for hydrodynamics to offer a successful and accurate description of experimental results. However, I predict the breakdown of hydrodynamics at momenta of order seven times the temperature, corresponding to a smallest possible QCD liquid drop size of 0.15 fm.
Keywords
Effective Field Theory Quasinormal Mode Nuclear Collision Pressure Anisotropy Gradient Expansion1 Preface
The intention of this note is to provide an impulse to the nuclear collisions community which in my opinion is trying to reconcile the traditional paradigm of hydrodynamic applicability with the unreasonable success of hydrodynamics at seemingly describing ever smaller systems. In an attempt of “outofthebox” thinking, my hope is to start a discussion as regards to the interpretation of this hydrodynamic success that challenges the traditional paradigm of local equilibration. Aiming at challenging the traditional, I found it very hard to conform to the standard manuscript style, which led to this informal note.
It should be pointed out that many of my main points have been made by others before me. All I have done is collect the available information and, based on this information, offer my own interpretations, conclusions, and predictions. I expect that these conclusions may appear obvious to some readers and contentious to others. I invite readers who do not agree with my statements to engage in a constructive dialog, which I hope could help us make progress.
2 Introduction
If nuclear collision experiments do not probe nearequilibrium matter, then this may have a number of consequences which to my knowledge have not been appreciated before, providing the motivation for this note. Firstly, it would imply that the nuclear experiments do not (directly) probe equilibrium QCD properties as those calculated in firstprinciple lattice QCD calculations. Depending on the degree of nonequilibrium, experiments may be closer to or farther away from the QCD phase diagram plane spanned by temperature and baryon chemical potential. While for hydrodynamics, a projection from nonequilibrium space to the equilibrium plane is provided by e.g. the Landau matching condition, for other observables such a projection is not explicitly known. For instance, it is possible that the phenomenon of critical fluctuations associated with the experimental search for a QCD critical point would get modified when experiments probe QCD away from equilibrium.
The understanding that the matter created in high energy nuclear collisions does not need to equilibrate or isotropize locally in order for hydrodynamics to be quantitatively applicable would imply that the “early thermalization puzzle” is to some extent not a genuine puzzle (see more details about this below). On the other hand, having experimental access to a nonequilibrium quantum system could lead to new directions in the field such as, e.g. observing nonequilibrium entropy production, properties of nonthermal fixed points or offequilibrium photoproduction.
Finally, if systems created in nuclear collisions do not equilibrate this could naturally explain why proton–proton collision data on azimuthal correlations appears to be so similar to data obtained in nucleusnucleus collisions. Once small gradients or nearequilibrium is no longer a requirement, hydrodynamics will generically convert initial state geometry and fluctuations into correlations, thus making large and small systems look alike in their azimuthal correlation signals. Pushing this idea even further would imply that any lump of sufficiently high energy density could expand according to the laws of hydrodynamics (with one important caveat which will be discussed below). A natural consequence of this would be the presence of exponentially falling (thermal) spectra as well as potential azimuthal correlations in \(\mathrm{e}^+\)+\(\mathrm{e}^\) collisions.
3 A historical perspective
In recent years, it has been demonstrated that experimental results obtained in relativistic nuclear collisions are well described by hydrodynamic simulations. Based on the paradigm that hydrodynamics requires nearequilibrium in order to be applicable, this successful hydrodynamic description has been interpreted as evidence for a locally equilibrated state of matter (dubbed the quark–gluon plasma) in high energy nuclear collisions.
The question on how the system created in high energy nuclear collisions reaches or at least comes close to equilibrium subsequently has led to a number of developments. In particular, it was realized that because of the expansion of the matter into the vacuum following the nuclear collision, the system would cool and thus freeze into a hadronic gas quickly, at which point the type of correlations observed in experiment could not longer be built up. Thus, it became apparent that a fluid dynamic approximation to the system dynamics had to start early, on a time scale of \(\tau \sim 1\) fm/c or less [1, 2, 3, 4].
Arnold et al. [7] pointed out that a possible way out of the dilemma was that full thermalization was actually not required for a hydrodynamic description, and that local (near) isotropy of the pressure tensor was sufficient. Thus, the attention of the field shifted toward finding a mechanism to quickly achieve local isotropy (isotropization) rather than full thermalization in high energy nuclear collisions.
One such possible mechanism was that of nonabelian plasma instabilities, specifically the nonabelian Weibel instability, which had been extensively studied by Mrowczynski since the 1980s [8, 9]. In a series of numerical studies by a number of groups the growth and saturation of these plasma instabilities was determined for nonexpanding systems [10, 11, 12, 13, 14], see Refs. [15, 16] for a review. While corroborating the initial exponential approach toward isotropy, these numerical studies suggested the system to stall at large pressure anisotropies once the plasma instabilities reached the nonperturbative nonabelian scale and could no longer grow exponentially. Even worse, later studies in expanding systems aiming at more realistically describing experimental nuclear collisions indicated that the effect of plasma instabilities was delayed/diminished to an extent that they could not lead to local pressure isotropy in a time scale relevant for nuclear collisions at RHIC and the LHC [17, 18, 19, 20].
While full isotropization seemed difficult to achieve within a weakcoupling QCD based framework, it appeared to be reachable much faster in gravitational duals of gauge theories in the limit of infinite coupling. For instance, Chesler and Yaffe [21] report isotropization to occur at \(\tau \sim 0.7/T\) for a nonexpanding system, roughly translating to \(\tau \simeq 0.35\) fm/c when assuming \(T\sim 0.4\) GeV. However, similar to the case of plasma instabilities, isotropization does take more time when considering the case of expanding systems such as those in nuclear collisions, because expansion constantly tries to drive the system away from local isotropy. This is the reason why in newer studies including expansion [22, 23], the isotropization time gets delayed. In particular, it eventually became clear that even for systems at infinite coupling strength the system does not isotropize early. Rather, even at infinite coupling, the pressure anisotropy exceeds 10 percent for all times \(\tau \lesssim 10\) fm/c [24].
For completeness, it should be noted that when including inelastic processes in a weakcoupling based descriptions, recent studies [25, 26] have demonstrated the approach to isotropy, albeit at times later than those found for infinitely strongly coupled gauge theories. (This would better be the case). Thus, the approach to isotropization in expanding gauge theories is now understood both at weak and strong coupling, and indicates long times.
Despite the impressive progress made, I believe it is a correct statement to say that at phenomenologically relevant times of \(\tau \sim 1\) fm/c following a nuclear collision, no theoretical approach (be it weakly coupled or strongly coupled) finds the longitudinal and transverse pressure to agree with each other to better than a factor of two. Obviously, a pressure anisotropy of 50 percent is not close to an isotropic system, let alone a system in thermal equilibrium. By the criterion of Arnold, Lenaghan, Moore and Yaffe, hydrodynamics should not apply.
But it does.
4 Hydrodynamization or the onset of hydrodynamic applicability

Q: How do you people know hydrodynamics applies for pressure anisotropies of 50 percent or larger?

A: We checked.
Let us consider the following numerical experiment. Take matter described by gauge/gravity duality at infinite coupling or alternatively described by kinetic theory at some finite (constant) value of the coupling. Let the matter be initially at rest in equilibrium with some temperature \(T_i\) in flat Minkowski space. Then, at a time \(t\sim 0\), the spacetime suddenly starts to expand in one dimension so that it effectively mimics the effects of socalled Bjorken flow [27]. The symbols in Fig. 1 show the response of the matter (at various values of the coupling \(\lambda \)) in terms of the ratio of longitudinal to transverse pressure as a function of time. The matter is initially in equilibrium so \(P_T=P_L\) (zero pressure anisotropy) and also tends to equilibrium at late times when the expansion becomes very slow. However, for \(t\simeq 0\) when the expansion is most rapid, the matter is clearly not locally isotropic, and deviations from local isotropy become larger as the coupling is decreased.
Also shown in Fig. 1 are results from a hydrodynamic gradient expansion to first and second order in gradients, respectively. One observes that hydrodynamics quantitatively matches the exact results whenever the pressure anisotropy is 50 percent or smaller.
This ‘unreasonable success’ of hydrodynamics in describing systems with pressure anisotropies of order unity is neither limited to this one example nor to AdS/CFT dynamics, nor exclusively to previous work by the present author, cf. Refs. [23, 28, 29, 30, 31, 32].
While of course no general proof, the above numerical experiment indicates that hydrodynamics is able to give quantitatively accurate descriptions even when the matter not locally isotropic. The time scale at which hydrodynamics first is able to closely approximate the subsequent dynamics of the exact underlying microscopic theory has been dubbed hydrodynamization time [33]. At the hydrodynamization time, the matter is typically not locally isotropic. So what sets the time scale for the onset of the applicability of hydrodynamics?
5 Hydrodynamic versus nonhydrodynamic modes
What is hydrodynamics? The equations of hydrodynamics can be derived using a multitude of approaches. Some assume the system to be close to thermal equilibrium, others assume a weakly coupled microscopic particle description (kinetic theory). In my opinion, the most general derivation of hydrodynamics follows the approach of effective field theory (EFT).
According to this viewpoint, hydrodynamics is the EFT of longlived, longwavelength excitations consistent with the basic symmetries of the underlying system. The fundamental variables of the EFT are that of a fluid: pressure P, (energy) density \(\epsilon \), and fluid velocity \(u^a\). To lowest (leading) order in the EFT, only algebraic combinations of these quantities will enter the description.^{1} Corrections can then be systematically obtained by considering gradients of the fundamental variables.
The above EFT derivation does at no point invoke the presence of an underlying particlebased, kinetic description of the matter. However, given the requirement of the small gradients, it does seem to require the system to be close to isotropy. So what if gradients were not small in a particular situation of interest? Obviously, stopping at first order in a gradient expansion would not be a good approximation. However, one could try to include higherorder gradient corrections to obtain a good approximation. I will try to elucidate what happens in this case through a particular example.
If the gradient expansion was convergent, then we would have succeeded in a (highorder) theory of hydrodynamics that was unconditionally applicable also when the gradients are large. Given that for this theory a very large number of coefficients \(\alpha _n\) had to be calculated, it would be cumbersome if not impossible to generalize this approach to situations with a much lower degree of symmetry (e.g. nuclear collisions), but at least in principle, it would work!
Unfortunately, there is mounting evidence that the hydrodynamic gradient expansion generally is not a convergent series. In the cases that have been examined in detail (\(\mathcal{N}=4\) and \(\mathcal{N}=2^*\) SYM at infinite coupling, weakly coupled kinetic theory in the relaxation time approximation and Müller–Israel–Stewart (MIS) theory) it was found that \(\alpha _n\propto n!\) for large n, thus making the gradient expansion a divergent series [36, 37, 38, 39, 40].
There are two things to note about the resummed result (4). First, the exponential multiplying the coefficient \(\gamma \) in (4) cannot be recast in terms of the hydrodynamic gradient expansion. It is a truly nonhydrodynamic mode, and its presence explains why the naive hydrodynamic gradient series is divergent.
Contrary to the hydrodynamic contribution, the nonhydrodynamic contribution \(T^{ab}_\mathrm{non{\text{ }}hydro}\) will in general not have a universal form, but rather be dependent on the particular underlying microscopic description under consideration (“microscopic” in the sense of QCD, not in the sense of quasiparticles).
More important than the realization that the energymomentum tensor can be split into a hydrodynamic and a nonhydrodynamic piece is the fact that the hydrodynamic poles cease to exist at some value of \(\mathbf{k}\) when the coupling is finite [48]. This implies that, for \(\mathbf{k}\) larger than some critical value of \(\mathbf{k}_c\) (dependent on the coupling), the hydrodynamic component vanishes from the spectrum and is replaced by purely nonhydrodynamic behavior. At least for \(\mathbf{k}>\mathbf{k}_c\), hydrodynamics has broken down.
One may criticize that \(\mathcal{N}=4\) SYM is a very special microscopic theory, and worry about drawing general conclusions based exclusively on \(\mathcal{N}=4\) SYM. However, it turns out that when calculating \(G_R^{ab,cd}(\omega ,\mathbf{k})\) in kinetic theory in the relaxation time approximation [46], similar conclusions apply. In kinetic theory, \(G_R^{00,00}(\omega ,\mathbf{k})\) generally has two hydrodynamic poles which are located at \(\omega _h=\pm c_s \mathbf{k} \frac{2 i \eta \mathbf{k}^2}{3 s}\) when \(\mathbf{k}\ll 1\). In addition to these hydrodynamic poles, \(G_R^{00,00}(\omega ,\mathbf{k})\) exhibits a logarithmic branch cut which may be taken to run from \(\mathbf{k}\frac{i}{\tau _R}\) to \(\mathbf{k}\frac{i}{\tau _R}\) where \(\tau _R=5 \frac{\eta }{s T}\) is the relaxation time in kinetic theory. It is interesting to note that when increasing \(\mathbf{k}\) beyond some critical value \(\mathbf{k}_c\), the hydrodynamic poles pass through the logarithmic cut onto the next Riemann sheet, and effectively cease to exist (see Fig. 2). Only the nonhydrodynamic branch cut remains, implying that, for \(\mathbf{k}>\mathbf{k}_c\), hydrodynamics has broken down.
I will summarize the above observations in the form of a
Lemma
Given the existence of a local rest frame, hydrodynamics offers a valid and quantitatively reliable description of the energymomentum tensor even in nonequilibrium situations as long as the contribution from all nonhydrodynamic modes can be neglected.
Proof
Consider matter possessing a local rest frame everywhere in the spacetime patch of interest, such that the local energy density is nonnegative in any frame. Pick a convenient global frame (“laboratory frame”) and consider the Fourier decomposition of the energymomentum tensor in this frame. Now consider real time perturbations \(\delta T^{ab}(t,\mathbf{k})\) around the Fourier zero mode \(T^{ab}_\mathrm{background}\) in the laboratory frame. At some initial time \(t_0\), the difference between the local energymomentum tensor and the background can be viewed as an initial perturbation \(S^{ab}(t_0,\mathbf{k})\). In the limit of small perturbation amplitude \(S^{ab}\rightarrow 0\), linear response theory applies, cf. Eq. (7). Furthermore, the retarded twopoint function \(G_R\) is well known to be given by Navier–Stokes hydrodynamics [45] in the small wavenumber limit \(k\rightarrow 0\). The twopoint correlator in Navier–Stokes hydrodynamics possesses hydrodynamic poles (shear and sound poles) in the complex frequency plane. Contour integration as in Eq. (7) will pick up these poles and lead to a hydrodynamic contribution to \(\delta T^{ab}\). As k is increased, the location of the hydrodynamic poles may shift, and they may even disappear from the spectrum completely at some critical wavenumber. In addition to the hydrodynamic poles, new, nonhydrodynamic singularities may appear in the complex frequency plane. These nonhydrodynamic singularities, upon contour integration in Eq. (7) will lead to a nonhydrodynamic contribution that has to be added to the hydrodynamic part of \(\delta T^{ab}\) as in the example given in Eq. (8). As the amplitude \(S^{ab}\) is increased, other, nonlinear structures will contribute to \(\delta T^{ab}\) which can be expressed as a sum over integrals of npoint functions with the appropriate powers of the source \(S^{ab}\). In the limit of small wavenumber, these nonlinear corrections to the hydrodynamic part will, upon resummation, shift the hydrodynamic poles locally, and in addition contribute new structures familiar from the hydrodynamic gradient expansion, cf. Eq. (2). Away from \(k=0\) the nonlinear hydrodynamic part of \(\delta T^{ab}\) is analytically connected to this familiar structure, giving rise to the generalized hydrodynamic attractor form, until some critical wavenumber \(k=k_c\) is reached. In addition to the hydrodynamic part, there will be nonlinear corrections to the nonhydrodynamic contribution of \(\delta T^{ab}\). Thus, the global energymomentum tensor can be written as \(T^{ab}=T^{ab}_\mathrm{background}+\delta T^{ab}=T^{ab}_\mathrm{hydro}+T^{ab}_\mathrm{non{\text{ }}hydro}\). If the nonhydrodynamic contribution can be neglected, the lemma follows trivially.
(Mathematicians probably would want to see a more formal proof than this, so the above lemma should probably be called a conjecture).
The above lemma may seem trivial: once nonhydrodynamic modes are absent, how can the energymomentum tensor be described by anything else but hydrodynamics? However, when phrased in this fashion, hydrodynamics neither requires equilibrium, nor isotropy, nor infinitesimally small gradients. (It does require the presence of a local rest frame, though [34]). The applicability of hydrodynamics is exclusively determined by the relative importance of nonhydrodynamic modes.
As it stands, the above lemma has at least one potentially important consequence, which is phrased as follows.
Dilemma
The phenomenological success of hydrodynamics in describing experimental data from high energy nuclear collisions does not imply nearequilibrium behavior of the matter. Experiments do not directly probe the equilibrium QCD phase diagram at finite \(T,\mu \), but explore trajectories in a space with at least one more (nonequilibrium) direction.
Incomplete equilibration in nuclear collisions is a fact well known to all heavyion hydro practitioners in the field and has been pointed out more than a decade ago by Bhalerao, Blaizot, Borghini and Ollitrault in Ref. [49]. The above lemma allows me to go one step further since not even nearequilibrium is required for hydrodynamics. The second part of the dilemma is a direct consequence of the first, yet it probably is not as widely appreciated. I have tried to visualize the last point of the dilemma in Fig. 3. To generate the hypothetical trajectories I have matched the pressure anisotropy \(P_L/P_T\) to the momentum anisotropy parameter \(\xi \), first defined in Ref. [50]. I use \(\xi \, \in [0,\infty )\) to express the degree of nonequilibrium (“nonequilibriumness”), where \(\xi =0\) corresponds to the case of local equilibrium. There is an extensive literature in anisotropic hydrodynamics which makes the connection between \(P_L/P_T\) and \(\xi \) precise (see e.g. the instructive lecture notes by Strickland, Ref. [51, Eq. (3.19)]). For the pressure anisotropy itself, I used Navier–Stokes hydrodynamics in Bjorken expansion (cf. [24]) and a viscosity that is given by \(\frac{\eta }{s}=\frac{1}{4\pi }\) for \(T>0.17\) GeV and rises linearly as the temperature is lowered to reach \(\frac{\eta }{s}=1\) at \(T=0.1\) GeV (cf. [52, 53]). The temperature and chemical potential dependence result from a drastically simplified version of Hung and Shuryak, Ref. [54], with just one massive degree of freedom in the hadron gas phase, and assuming constant baryon density to entropy ratio. Trajectories are started at \(\tau =1\) fm/c with multiplicities representative of experimental measurements [55] converted to temperature values \(T(\tau =1\,\mathrm{fm/c})\) as in Ref. [56]. Clearly, none of these choices do justice to the much more accurate descriptions that are currently available. However, I expect the sketch in Fig. 3 to be qualitatively reliable.
6 Breakdown of hydrodynamics and tests
According to the central lemma in the previous section, hydrodynamics can be used to describe a system if a local rest frame exists and nonhydrodynamic modes are subdominant. I have nothing new to say about how to test for the presence of a local rest frame, so I will simply follow everyone else’s approach and assume that a local rest frame exists. (This is very likely wrong [34] and actually should be thought about more, but I leave this task to dedicated readers.)
Naively applying the results for \(k_c\) to QCD with \(\alpha _s\equiv \frac{g^2}{4\pi }\simeq 0.5\) leads to \(\frac{k_c(\lambda \simeq 19)}{T}=47\), where the higher value is actually coming from the kinetic theory result. Let us be optimistic and take \(\mathbf{k}_c \simeq 7 T\). Thus, I claim that no hydrodynamic description is possible for QCD systems smaller than \(k_c^{1}\simeq 0.15\) fm at a typical QCDscale temperature of \(T\simeq 200\) MeV. The prediction that hydrodynamics must break down for \(\mathbf{k}^{1}<0.15\) fm is most likely somewhat useless, because it is very hard to falsify experimentally in nuclear collisions given that the mean proton radius is 0.86 fm. However, the limit of 0.15 fm at least constitutes an actual numerical conjecture for the lower bound of the smallest possible droplet of QCD liquid.
A much more direct route to experimentally constrain \(k_c\) could be provided by highmomentum data on flow coefficients, see Fig. 5. Experimental data for collective flow harmonics in Pb+Pb collisions suggests a change in behavior in the regime between \(p_T=3\) GeV to \(p_T=4\) GeV. The lowmomentum region is well described by hydrodynamics [60]. Assuming that measured particles originated from a constanttemperature freezeout surface at \(T=0.17\) GeV, this would indicate a breakdown of hydrodynamic behavior at \(\frac{p_T}{T}=1823\). In order to relate this scale to the hydrodynamic breakdown scale \(k_c\lesssim 7 T\), quantitative calculations of the location of nonhydrodynamic modes in an expanding system are needed.
In the hadronic phase, kinetic theory would predict hydrodynamic modes to dominate for \(k<k_c\propto \frac{1}{\tau _R}\), while nonhydrodynamic (particle) modes dominate for \(k>k_c\). As the temperature is lowered, \(\tau _R\propto \frac{\eta }{s T}\) increases strongly [52] until \(k_c\) falls below the typical system wavenumber. From this point onward, most of the system dynamics proceeds according to the nonhydrodynamic particle kinetics, providing a qualitative understanding of the transition from hydrodynamic to particle cascade dynamics (“freezeout”).
The above statements involve hard lower bounds on the smallest scales at which hydrodynamics applies, and a qualitative understanding of the freezeout transition. However, a quantitative test of the applicability of hydrodynamics, e.g. through testing sensitivity of results with respect to nonhydrodynamic modes, is desirable in the case of nuclear collisions. Fortunately, the workhorse of relativistic viscous hydrodynamics simulations, “causal relativistic viscous hydrodynamics” (which goes by many names and acronyms but is usually associated with the work of Müller, Israel and Stewart [61, 62]) does contain a nonhydrodynamic mode buried within, which may be exploited for testing purposes. Specifically, besides the usual hydrodynamic modes, the energymomentum tensor twopoint function contains a pole located at \(\omega _{nh}=\frac{i}{\tau _\pi }\), where \(\tau _\pi \) is the “viscous relaxation time” that also controls the size of the secondorder gradient term in the onepoint function of \(T^{ab}\). Any current numerical hydrodynamics simulation of the matter produced after a relativistic nuclear collision needs a specific value for \(\tau _\pi \) as an input. Simulators choose values of \(\tau _\pi \) as they see fit, given that the “correct” value for \(\tau _\pi \) for QCD is not known, and that primary interest is in extracting information as regards \(\frac{\eta }{s},\frac{\zeta }{s}\), not some obscure secondorder transport coefficient.
However, varying \(\tau _\pi \) around some “fiducial” value does vary the decay time of the nonhydrodynamic mode inherent to causal relativistic hydrodynamics, thus offering a direct handle on the sensitivity of final results on the nonhydrodynamic mode. This can be implemented in practice in relativistic viscous hydrodynamic simulations by running simulations at multiple values of \(\tau _\pi \) and expressing final results in terms of a mean value and a systematic error bar covering the variations of final results from changing \(\tau _\pi \). Examples are shown in Fig. 6 for the case of central p+Au collisions at various values of \(\sqrt{s}\) and p+p collisions at \(\sqrt{s}=7\) TeV. While the sensitivity on nonhydrodynamic modes is not vanishingly small, the error bars do seem to signal the applicability of hydrodynamics to both p+Au and p+p collisions in general. However, in the case of p+p collisions and \(\frac{\mathrm{d}N}{\mathrm{d}Y}<2\), the error bars become large, signaling strong sensitivity of the result to nonhydrodynamic modes. This empirical result seems to indicate that hydrodynamics breaks down in p+p collisions for \(\frac{\mathrm{d}N}{\mathrm{d}Y}<2\). This multiplicity value of the hydrodynamic breakdown corresponds well to the results derived by Spalinski [63]. It would be interesting to repeat these sensitivity tests for hydrodynamics with a different nonhydrodynamic mode structure, for instance along the lines suggested in Ref. [64].
7 Concluding remarks
 1.
I would argue that there is hard experimental evidence, e.g. through the phenomenon of jet modifications, for the presence of strongly interacting QCD matter created in nuclear collisions. As argued in this note, I am doubtful about the hard evidence for this matter to be equilibrated.
 2.
The physics of nonhydrodynamic modes is a rich and a barely studied subject. Given that nonhydrodynamic modes play an important role in the applicability and breakdown of a hydrodynamic descriptions, I believe those nonhydro modes should receive more attention, from theorists and experimentalists alike.
 3.
The central lemma in Sect. 5 also applies to the case of diffusion, not only momentum transport. In particular, this implies that a constitutive equation of the form \(J=\sigma E\) could hold in the earlytime, outofequilibrium regime following a nuclear collision, if nonhydro modes are subdominant. This could potentially explain a longerthanexpected lifetime of the magnetic field which is critical to experimental detection of the Chiral Magnetic Effect [67] (see also Ref. [68]).
 4.
As outlined in the central dilemma in Sect. 5, the experimental search for the QCD critical point will necessarily explore trajectories in some nonequilibrium space (cf. Fig. 3). This implies that the standard equilibrium theory of critical fluctuations strictly speaking does not apply, and one should try to understand nonequilibrium effects (see e.g. Ref. [69]) in order to correctly interpret the experimental data.
 5.
In view of the ‘QGP drop size lower bound’ of 0.15 fm, it is maybe not surprising that the matter created in p+p collisions would behave hydrodynamically. At this scale, however, p+p collisions may not be the ultimate drop size test. QCDQED couplings allow fluctuations of electrons to, e.g. quark pairs, thus opening up the possibility of local energy deposition reminiscent of p+p collisions occurring in \(\mathrm{e}^+\)–\(\mathrm{e}^\) collisions (cf. Refs. [70, 71, 72]). Data on \(\mathrm{e}^+\)–\(\mathrm{e}^\) collisions taken at, e.g. LEP should be reanalyzed with modern tools in order to find (or rule out) hydrodynamic behavior in these systems.
 6.
The fact that experimental data shows a qualitative change in trend from hydrodynamic behavior at low momenta to nonhydrodynamic behavior at high momenta suggests a potential experimental handle on the hydrodynamic breakdown scale \(k_c\) in QCD. This potential connection should be made quantitative in further studies.
 7.
The entire discussion in this note ignores the presence of hydrodynamic thermal fluctuations, which arise in SU(N) gauge theories at any finite number N. The subfield of relativistic hydrodynamics with thermal fluctuations is still in its infancy, but potentially can have important phenomenological consequences [45, 73, 74, 75, 76, 77, 78, 79, 80].
 8.
A recurring problem of nonstandard cosmology (socalled “viscous cosmology”, cf. [81, 82]) seems to be that “interesting” deviations from standard cosmology occur when gradient corrections become order unity. In the “oldfashioned” picture of hydrodynamics, this was not acceptable since order unity corrections heralded the breakdown of applicability of the theory. In view of the central lemma in Sect. 5, it could be interesting to determine the relevant nonhydrodynamic modes in cosmology and reevaluate the regime of applicability of viscous cosmologies.
Footnotes
Notes
Acknowledgements
I am indebted to many colleagues for countless fruitful discussions on the topics mentioned in this note. In particular I would like to thank J. CassalderreySolana, T. DeGrand M. Floris, J. Nagle, A. Kurkela, J. Schukraft, M. Spalinski, A. Starinets, M. Stephanov, and W. van der Schee for discussions, and the organizers of the CERNtheory workshop “The Big Bang and the little bangs”, the Oxford workshop on “NonEquilibrium Physics and Holography”, the Santiago de Compostela meeting on “Numerical Relativity and Holography” and the Technion workshop on “Numerical Methods for AdS spaces” for providing interesting meetings and many stimulating discussions. Moreover, I would like to thank F. Becattini, M. Heller, S. Morwczynski, J. Nagle (again!), J. Noronha, J. Schukraft (again!), L. Yaffe and B. Zajc for their detailed and encouraging comments on the first version of this note and S. Grozdanov, N. Kaplis and A. Starinets for providing their exact numerical data on \(k_c\) from Ref. [48]. I would like to thank the US Department of Energy for providing financial support under DOE Award No. DESC0008132. Finally, I am grateful to Lufthansa for many hours without phone or internet, leaving me no other choice than to think about physics.
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