Relativistic Hydrodynamic Attractors with Broken Symmetries: Non-Conformal and Non-Homogeneous
PPrepared for submission to JHEP
Relativistic Hydrodynamic Attractors with
Broken Symmetries: Non-Conformal and
Non-Homogeneous
Paul Romatschke , Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309,USA
E-mail: [email protected]
Abstract:
Standard textbooks will state that hydrodynamics requires near-equilibriumto be applicable. Recently, however, out-of-equilibrium attractor solutions for hydro-dynamics have been found in kinetic theory and holography in systems with a highdegree of symmetry, suggesting the possibility of a genuine out-of-equilibrium formu-lation of hydrodynamics. This work demonstrates that attractor solutions also occurin non-conformal kinetic theory and spatially non-homogeneous systems, potentiallyhaving important implications for the interpretation of experimental data in heavy-ionand proton-proton collisions and relativistic fluid dynamics as a whole. a r X i v : . [ h e p - t h ] J a n ontents Recently, attractor solutions for relativistic hydrodynamics have been found in varioussystems with a high degree of symmetry [1–4]. Besides their relation to the mathemat-ical theory of resurgence [5, 6], these attractor solutions are interesting because theyimply a firm theoretical foundation for the applicability of hydrodynamics in out-of-equilibrium situations [2, 7–9]. A key question in this context is if attractor solutionscan be found in systems that do not exhibit additional symmetries.A central property of hydrodynamic attractor solutions is that they become in-distinguishable from solutions of relativistic dissipative hydrodynamics in the limit ofsmall gradients. As such it is useful to first consider the relativistic generalization ofthe Navier-Stokes equations [12] D(cid:15) + ( (cid:15) + P ) ∇ λ u λ = η σ µν σ µν + ζ (cid:0) ∇ λ u λ (cid:1) , (1.1)( (cid:15) + P ) Du α + ∇ α ⊥ P = ∆ αν ∇ µ (cid:0) ησ µν + ζ ∆ µν ∇ λ u λ (cid:1) , (1.2)where (cid:15), P, u µ are the fluid’s local energy density, pressure and velocity, and ∇ µ isthe geometric covariant derivative. Furthermore, η, ζ are the shear and bulk viscositycoefficients, D ≡ u µ ∇ µ , ∇ µ ⊥ ≡ ∆ µν ∇ ν , ∆ µν ≡ g µν + u µ u ν , σ µν ≡ ∇ µ ⊥ u ν + ∇ ν ⊥ u µ − ∆ µν ∇ µ u µ and the mostly plus convention for the metric tensor g µν will be used. For the purpose of this discussion, the well-known problems of acausality and instability of theNavier-Stokes equations [10, 11] can be safely ignored. – 1 –n the absence of any conserved charges, the Navier-Stokes equation for the energy-density evolution may be rewritten as
D(cid:15) ( (cid:15) + P ) ∇ λ u λ = D ln s ∇ λ u λ = − η s σ µν σ µν T ∇ λ u λ + ζs ∇ λ u λ T , (1.3)where s, T are the entropy density and temperature, respectively, related to (cid:15), P via theusual thermodynamic relations. For relativistic systems, which are never incompress-ible, the expansion scalar ∇ λ u λ is generally non-vanishing unless global equilibriumis reached. Eq. (1.3) then implies that for a given system, the time-evolution of thequantity A = D ln s ∇ λ u λ , (1.4)will behave similarly for small gradients irrespective of initial conditions. As an ex-ample, consider systems exhibiting conformal symmetry ζ = 0 and ηs = const. In thiscase, Eq. (1.3) implies that the time evolution of D ln s ∇ λ u λ for arbitrary initial conditionswill collapse onto a single curve when expressed as a function of sT ∇ λ u λ ησ µν σ µν .As another example, consider non-conformal systems where the trace of the energy-momentum tensor of the Navier-Stokes equation is given by Tr T µν = − (cid:15) +3 P − ζ ∇ λ u λ .The trace of the energy-momentum tensor corresponds to the sum of temporal andspatial eigenvalues, and thus encodes information about the effective equation of statethe system is experiencing. In equilibrium, ∇ λ u λ = 0, and hence the trace anomalyimplies an equilibrium equation of state. For non-conformal systems out of equilibrium,the Navier-Stokes equation implies that the time-evolution of the quantity A = Tr T µν + (cid:15) − PζT , (1.5)will behave similarly for small gradients irrespective of initial conditions if expressedas a function of the inverse gradient strengthΓ ≡ (cid:20) η s σ µν σ µν T ∇ λ u λ + ζs ∇ λ u λ T (cid:21) − . (1.6)Clearly, Γ reduces to the expression for the scaling strength found in the conformalcase ζ = 0 above.Real systems exhibit deviations from the behavior predicted by the Navier-Stokesequation at finite gradient strength. Nevertheless, solutions will tend to the Navier-Stokes solution in the limit of small gradients, such that it acts as an attractor solution.Less trivial is the question of whether such an attractor solution extends to the regime– 2 –f moderate or even large gradients. In practice, one can search for attractor solu-tions in real systems or microscopic theories by e.g. studying the time-evolution ofquantities such as A , A . For A , this has been successfully achieved in system withconformal symmetry and restricted spatial dynamics (Bjorken flow in the longitudinaldirection[13] and spatially homogeneous in the transverse directions) in Refs. [1–4]. Thepresent work is trying to extend the understanding of hydrodynamic attractor solutionsby studying non-conformal systems and systems that allow for spatial dynamics. Consider a gas of particles with mass m undergoing one-dimensional boost-invariantexpansion according to Bjorken [13]. It is convenient to work in Milne coordinatesproper time τ = √ t − z and spacetime rapidity arctanh( z/t ) for this system. Kinetictheory in the relaxation-time approximation is defined by a single-particle on-shelldistribution function f ( x µ , p µ ) which obeys the Boltzmann BGK equation [14, 15] p µ ∂ µ f − p λ p σ Γ µλσ ∂f∂p µ = p µ u µ ( f − f eq ) τ R , (2.1)where Γ µλσ are the Christoffel symbols for Milne coordinates and f eq = 2 π e p µ u µ /T . Inthese expressions, u µ ( x µ ) , T ( x µ ) are related to the time-like eigenvector and eigenvalueof the energy-momentum tensor T µν ( x µ ) = (cid:82) dχp µ p ν f as (cid:15)u µ ≡ − T µν u ν , (2.2)with the normalization condition u µ u µ = −
1. Here (cid:15) can be recognized as the localenergy density. For a massive gas at temperature T in equilibrium, the relation betweenenergy density and temperature is readily calculated from (2.2) with f = f eq . Workingin units where the particle mass m = 1, one finds [16] (cid:15) ( T ) = 3 T K ( T − ) + T K ( T − ) , P ( T ) = T K ( T − ) (2.3) s ( T ) = K ( T − ) , c s ( T ) = (cid:18) T − K ( T − ) K ( T − ) (cid:19) − (2.4) η ( T ) = τ R T (cid:82) T dT (cid:48) T (cid:48) s ( T (cid:48) ) , ζ ( T ) = η ( T ) − τ R T s ( T ) c s ( T ) (2.5)for the energy density, pressure P , entropy density s , speed of sound squared c s as wellas shear and bulk viscosity coefficients, respectively. Here K ( x ) denote modified Besselfunctions. Note that the integration measure is given by dχ ≡ d p (2 π ) (cid:112) − det g µν p )(2 π ) δ (cid:0) − g µν p µ p ν − m (cid:1) . – 3 –ut of equilibrium, there is no temperature T for the system, but there alwaysis an energy density that can be found from Eq. (2.2). To find the parameter T ( (cid:15) )in f eq , note that integration (cid:82) dχp ν of Eq. (2.1) leads to covariant conservation of theenergy-momentum tensor iff u µ T µν = u µ T µν eq [16]. Therefore, T ( (cid:15) ) out of equilibriumis required to be chosen such that the energy density in (2.3) matches the time-likeeigenvalue (cid:15) of T µν . Note that this is not implying that the system evolves with anequilibrium equation of state, which is a relation between the time-like and space-likeeigenvalues of T µν , but rather only fixing the setup of the equation (2.1).In a system that is homogeneous with respect to transverse coordinates x ⊥ = ( x, y )and boost-invariant (independent of space-time rapidity), Eq. (2.1) implies the followingintegral equation for the evolution of the energy-density as a function of proper time[17]: (cid:15) ( T ( τ )) = Λ D ( τ, τ ) h (cid:18) τ τ √ ξ , Λ − (cid:19) + (cid:90) ττ dτ (cid:48) τ R D ( τ, τ (cid:48) ) h (cid:18) τ (cid:48) τ , T − ( τ (cid:48) ) (cid:19) , (2.6) h ( y, z ) = y (cid:90) ∞ duu e −√ u + z (cid:32)(cid:112) y + z /u + 1 + z /u (cid:112) y − (cid:115) y − y + z /u (cid:33) ,D ( τ , τ ) = e − (cid:82) τ τ dτ (cid:48) τR . For simplicity, τ R = C π /T with constant C π was chosen in the following. Initial con-ditions for the system are characterized by choosing a value of ξ ∈ [ − , ∞ ) and Λ at τ = τ . Numerical solutions to Eq. (2.6) for Λ = 1 and various values of ξ can begenerated by the methods outlined in Ref. [18]. For a spatially homogeneous and boost-invariant system, Eq. (1.4) becomes A ( τ ) = τ ∂ τ ln s and T ∇ λ u λ σ µν σ µν = τ T , ∇ λ u λ T = τ T lead to Γ = τ /γ s with γ s ≡ η(cid:15) + P + ζ(cid:15) + P the temperature-dependent sound attenuationlength.The results from numerically solving Eq. (2.6) are shown in Fig. 1, along with theresults from the Navier-Stokes equation. One observes that for arbitrary initial choicesof ξ at fixed τ , the subsequent evolution tend to cluster in special ’attractor solutions’which eventually merge with the Navier-Stokes results. In the case of A , it is possibleto find points close to the attractor solution by employing the technique outlined in[2], namely the ’slow-roll’ condition ∂ τ A | τ = τ ,ξ = ξ = 0 [1]. For the case at hand, thiscondition becomes( (cid:15) + P ) ( ∂ τ (cid:15) + τ ∂ τ (cid:15) ) − τ ( ∂ τ (cid:15) ) (1 + c s )( (cid:15) + P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = τ ,ξ = ξ = 0 , (2.7)– 4 – igure 1 . Scaling variables A , A for non-conformal kinetic theory in Bjorken flow withΛ = 1, τ R = 0 . /T as a function of inverse gradient strength Γ. Note that for A , scalingthe gradient strength with the sound attenuation length implies that the Navier-Stokes resultcorresponds to an area rather than a single curve. The curves labeled ’Attractor LO approx.’are generated by solving Eq. (2.7) for ξ ( τ ) and evaluating A , A using (2.8). where (cid:15) ( τ ) = Λ h (cid:16) √ ξ , Λ − (cid:17) , ∂ τ (cid:15) ( τ ) = − Λ τ √ ξ h (1 , (cid:16) √ ξ , Λ − (cid:17) , (2.8) ∂ τ (cid:15) ( τ ) = − τ R (cid:16) ∂ τ (cid:15) ( τ ) + ( (cid:15) + P ) τ (cid:17) + Λ (cid:32) h (2 , (cid:18) √ ξ , Λ − (cid:19) τ (1+ ξ ) + h (1 , (cid:18) √ ξ , Λ − (cid:19) τ √ ξ (cid:33) . Solving (2.7) numerically for τ R = . T and τ = 0 .
1, Λ = 1 (all in mass units m = 1)leads to ξ (cid:39) .
8. Using this value of ξ as initial condition, an attractor solution for A may be constructed numerically by solving Eq. (2.6). Unlike the case of conformaltheories, repeating the above procedure for different starting times t will lead to aslightly different attractor curve. This can be understood from the fact that, in the non-conformal case, A is not a simple function of τ T alone because there is an additionalmass scale to contend with. As a consequence, an envelope of attractor curves for A isshown in Fig. 1, suggesting that the attractor in the non-conformal case is an extendedobject. The width of the attractor envelope is related to the value of Λ − and I havechecked that the conformal (zero-width) attractor from Ref. [2] is recovered in the limitΛ → ∞ .Results for A are also shown in Fig. 1. One observes a clustering of trajectoriesfor arbitrary initial condition similar to A , again suggesting an attractor solution at Another useful relation is (cid:15) ( T ) = T h (1 , T − ). – 5 –arly times that is distinct from the Navier-Stokes result. However, the approach ofindividual trajectories to the A attractor solution seems to be slower than for A .The region labeled ’Attractor’ in Fig. 1 marks the area where different A attractorsolutions to Eq. (2.6) have merged. This demonstrates that there are special solutionsto Eq. (2.6) which are attractors for A and A simultaneously. All attractor solutions discussed so far where restricted to spatially homogeneous sys-tems, begging the question if attractor solutions survive if the spatial dynamics is notstrongly restricted. To study this question, consider the mock-microscopic theory ofresummed BRSSS (rBRSSS for short) in conformal symmetry, which is defined by anenergy momentum tensor T µν = (cid:15)u µ u ν + P ∆ µν + π µν with the dynamic shear stress π µν obeying the equations of motion [19] π µν = − ησ µν − τ π (cid:20) (cid:104) Dπ µν (cid:105) + 43 π µν ∇ ⊥ λ u λ (cid:21) + κ (cid:2) R <µν> − u λ u ρ R λ<µν>ρ (cid:3) + λ η π <µλ π ν>λ − λ η π <µλ Ω ν>λ + λ Ω <µλ Ω ν>λ . (3.1)In the following, only flat-space systems where the Ricci and Riemann tensors vanishare considered. Also, for simplicity I will set λ = λ = λ = 0. The rBRSSS equationsof motion are causal as long as τ π ≥ ηsT , cf. Ref. [20].For a spatially homogeneous system, the rBRSSS equations have been shown to pos-sess a hydrodynamic attractor solution [1]. Fortunately, numerical solvers for rBRSSSequations are readily available for spatially non-homogeneous systems [21–25]. I will beusing the VH2+1 solver from Ref. [21], which solves the rBRSSS equations for systemsthat are boost-invariant, but otherwise unrestricted in terms of the dynamics in trans-verse coordinates x ⊥ = ( x, y ). Using an optical Glauber model of a Au+Au collision atan impact parameter of 8 fm with AdS/CFT pre-equilibrium flow [26] as initial condi-tion , , the equations of motion are solved numerically on a lattice in transverse space x ⊥ . On each lattice point at each time-step, it is possible to evaluate A , Γ by locallycalculating (1.4),(1.6) numerically. A representative plot of the resulting trajectories isshown in Fig. 2. For the details of the implementation of the Glauber model see for instance Ref. [20]. In essence,the initial conditions considered here are qualitatively similar to a two-dimensional Gaussian energydensity (cid:15) ( τ = τ , x, y ) with different width in x, y . While it would have been possible to consider simpler initial conditions, the choice of Glauber+pre-equilibrium flow corresponds to studying attractors in the superSONIC model used in relativistic ioncollision phenomenology [27, 28]. – 6 – igure 2 . Selected trajectories of A (Γ) from solving rBRSSS equations numerically in2+1d for a Au+Au collision at b = 8 fm impact parameter with ηs = 0 . , τ π T = ηs . Thedirection of the time evolution is indicated by arrows on one trajectory. While at early times(not shown), trajectories are far separated and strongly dependent on initial conditions, atlate times one observes a clustering of trajectories near an apparent attractor solution. Theattractor solution does not stop near Γ (cid:39) , A (cid:39) − .
96, but system evolution of A (Γ)slows down dramatically in real time near this point, making it computationally expensive tocontinue tracking the attractor. As can be seen from this figure, trajectories in the A , Γ plane are initially farseparated, with some of the trajectories being close to the Navier-Stokes result, whileothers are not. However, at late times in the system evolution when gradients areno longer dominated solely by longitudinal Bjorken flow, trajectories cluster near anapparent attractor solution. Once the system comes close to the regime near Γ (cid:39) , A (cid:39) − .
96, evolution of A (Γ) slows down dramatically in real time, making itcomputationally expensive to continue tracking the attractor. I have checked thatthe apparent attractor remains unaffected by choosing different initial temperatures,starting times and impact parameters for the rBRSSS solution. Prior to the present study, relativistic hydrodynamic attractor solutions had been iden-tified in systems with a high degree of symmetry (conformal symmetry and spatial– 7 –omogeneity). In this work, the existence of attractor solutions for system with brokensymmetries (non-conformal and spatially non-homogeneous) was investigated.Findings: • In the case of non-conformal kinetic theory undergoing Bjorken flow, there ex-ists an attractor solution for the quantity A in Eq. (1.4) which is qualitativelysimilar to, but quantitatively different from, the known attractor solution of theconformal case [2]. • In the case of non-conformal kinetic theory undergoing Bjorken flow, points closeto the non-conformal attractor can be calculated by a ’slow-roll’ approximation(2.7). • In the case of non-conformal kinetic theory undergoing Bjorken flow, the non-conformal attractor solution for A also acts as an attractor for the quantity A ,which controls the non-equilibrium equation of state (the relation between energyand non-equilibrium pressure). • In the case of conformal non-homogeneous rBRSSS theory, there exists an at-tractor solution for the quantity A (Γ) with definitions (1.4), (1.6) that can beconstructed numerically. • In the case of conformal non-homogeneous rBRSSS theory, the attractor solu-tion A (Γ) is only partially known because the numerical evolution slows downdramatically.Interpretations: • Together with previous results on this subject, the present work strongly suggeststhat non-analytic attractor solutions for relativistic hydrodynamics exist in abroad class of theories regardless of the underlying symmetries. • Traditionally, relativistic hydrodynamics has been defined via a gradient expan-sion, with Euler equation, Navier-Stokes, BRSSS [19, 29] and Grozdanov-Kaplistheory [30] the respective complete 0 th , 1 st , 2 nd and 3 rd order realizations. How-ever, the hydrodynamic gradient series is expected to be divergent [1, 5, 6, 31, 32],calling into question the meaning of solutions to the hydrodynamic gradient se-ries for any non-vanishing gradient strength [8]. The existence of hydrodynamicattractor solutions provides this meaning and serves as the foundation for a new,yet to be elaborated, theory of hydrodynamics out-of-equilibrium [2, 9].– 8 – The existence of an apparent attractor for the non-equilibrium equation of statein Fig. 1b eliminates the last remnant of the standard textbook ’hydrodynamicsrequires equilibrium’ paradigm. While the relation between (cid:15), P is by constructioncontrolled by the thermodynamic results (2.3), the system experiences a non-equilibrium pressure P eff = P − ζ B ( (cid:15), ∇ ) ∇ λ u λ where ζ B is a non-analytic functionthat depends on both the energy density as well as the gradient strength (denotedformally as ∇ ), similar to the shear viscosity coefficient η B defined in Ref. [2].Though not interpreted in this fashion, non-equilibrium equations of state are nowroutinely used (and indeed required!) to provide precision fits of hydrodynamicmodels of relativistic ion collisions to experimental data [33, 34]. • Given the above findings, it can be considered reasonably likely that an attractorsolution for QCD exists in the context of relativistic ion collisions. This attrac-tor solution would result in ’hydrodynamic-like’ behavior of the system withoutany requirement of system equilibration. Therefore, it would naturally explainthe experimentally observed ’hydrodynamic-like’ signatures in relativistic heavy-ion and proton-proton collisions [35] and possibly even indicate hydrodynamicbehavior in electron-positron collisions [36]. • Besides the immediate application to the field of relativistic nuclear collisions, theexistence of hydrodynamic attractor solutions may have important implicationsfor relativistic fluid dynamics in general, e.g. by providing a firm foundation forviscous cosmologies [37–39].
Note Added in Proof
While this work was being reviewed, results for the non-conformal kinetic theory at-tractor were presented by a different group in Ref. [40]. The results in Ref. [40] appearto be in full agreement with this work.
I’d like to thank S. Jeon, J. Casalderrey-Solana and M. Spali´nksi for fruitful discussionson this topic and the organizers for the Initial Stages 2017 conference for providingsuch a nice and stimulating conference. Furthermore, I’d like to thank F. Becattini andS. Mrowczynski for challenging me to demonstrate the existence of attractor solutionsbeyond highly symmetric systems. This research was funded by the Department ofEnergy, award number DE-SC0017905. – 9 – eferences [1] M. P. Heller and M. Spali´nski,
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