PHYSICAL REVIEW D 97, 044041 (2018)

Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow

† ‡ Alireza Behtash,1,2,* C. N. Cruz-Camacho,3, and M. Martinez1, 1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA 2Kavli Institute for Theoretical Physics University of California, Santa Barbara, California 93106, USA 3Universidad Nacional de Colombia, Sede Bogotá, Facultad de Ciencias, Departamento de Física, Grupo de Física Nuclear, Carrera 45 No 26-85, Edificio Uriel Guti´errez, Bogotá D.C. C.P. 1101, Colombia

(Received 19 November 2017; published 26 February 2018)

The nonequilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed nonplanar and the basin of attraction is essentially three dimensional. The attractors of each hydrodynamical model are compared with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydro- dynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second-order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansions diverge and use resurgence techniques to perform the resummation of these divergences. We also comment on a possible link between the manifold of steepest descent paths in path integrals and the basin of attraction for the attractors via Lyapunov functions that opens a new horizon toward an effective field theory description of hydrodynamics. Our findings indicate that the reorganization of the expansion series carried out by anisotropic hydrodynamics resums the Knudsen and inverse Reynolds numbers to all orders and thus, it can be understood as an effective theory for the far-from-equilibrium fluid dynamics.

DOI: 10.1103/PhysRevD.97.044041

I. INTRODUCTION AND SUMMARY recent experimental results of pp collisions at high energies [2,3] have shown evidence of collective flow behavior Hydrodynamics is an effective theory which describes similar to the one observed in ultrarelativistic heavy-ion the long-wavelength and/or small-frequency phenomena of collisions. The experimental data measured in pp collisions physical systems. The fluid dynamical equations of motion can be described quantitatively by using hydrodynamical are derived by assuming that the mean free path is smaller models [4–6]. On the other hand, different theoretical toy than the typical size of the system [1]. The existence of a models for both weak and strong coupling [7–18] have large separation between the microscopic and macroscopic presented an overwhelming evidence that hydrodynamics scales can be reformulated as a small gradient expansion becomes valid even in nonequilibrium situations where around a background (usually the thermal equilibrium large gradients are present during the spacetime evolution state) which varies slowly. Thus, hydrodynamics would of the fluid. The success of hydrodynamical models to be invalid in far-from-equilibrium situations where the describe small systems as well as applying it to far-from- gradients of the hydrodynamical fields are large. This equilibrium situations calls for a better theoretical under- might be the case in systems of small size. However, standing of the foundations of hydrodynamics. For different strongly coupled theories it has been shown *[email protected] that the hydrodynamical gradient series expansion has zero † [email protected] radius of convergence and thus, it diverges [19]. On the ‡ [email protected] other hand, the divergent behavior of the hydrodynamical series expansion is also well known in weakly coupled Published by the American Physical Society under the terms of systems based on the Boltzmann equation for relativistic the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to and nonrelativistic systems [20 24]. A more detailed the author(s) and the published article’s title, journal citation, mathematical analysis of the origin of this divergence and DOI. Funded by SCOAP3. has unveiled the existence of a unique universal solution,

2470-0010=2018=97(4)=044041(23) 044041-1 Published by the American Physical Society BEHTASH, CRUZ-CAMACHO, and MARTINEZ PHYS. REV. D 97, 044041 (2018) the so-called attractor [25]. The attractor "solution" is any reparametrization of the variables, which is a signature intrinsically related to the mathematical theory of resur- of Gubser flow geometry governed by a true three- gence [20,26,27] and its details depend on the particular dimensional (3D) autonomous dynamical system as theory under consideration [28–37]. In simple terms, the opposed to the Bjorken model.1 This is made more precise attractor is a set of points in the phase space of the in the context of this 3D dynamical system, where the dynamical variables to which a family of solutions of an linearization problem is reviewed and a mathematically evolution equation merge after transients have died out. In rigorous definition of an attractor is given. We estimate the relativistic hydrodynamics it has been found in recent years shape of the basin of attraction by giving an approximate that for far-from-equilibrium initial conditions the trajec- Lyapunov function near the attracting fixed point. There we tories in the phase space merge quickly towards a non- touch upon an important link between the basin of attraction thermal attractor before the system reaches the full thermal and attractors with the manifold of steepest descent paths in equilibrium. This type of nonequilibrium attractor can be the path-integral formalism of quantum field theories and fully determined by very few terms of the gradient series of briefly discuss the role of a Lyapunov function as an relatively large size which involves transient nonhydrody- analogue of some effective action for the stable hydrody- namical degrees of freedom [19,20]. This property of the namical and nonhydrodynamical modes. attractor solution indicates that the system reaches its We finally discuss the properties of attractors of second- hydrodynamical behavior and thus, hydrodynamizes at order hydrodynamical theories [Israel-Stewart (IS) [44] and scales of time shorter than the typical thermalization and transient fluid (DNMR) theory [45]], anisotropic hydro- isotropization scales. The fact that hydrodynamization hap- dynamics (aHydro) [46–61] and the exact kinetic theory pens in different size systems on short scales of time while solution of the Gubser flow [9,13]. The numerical compar- exhibiting this degree of universality in far-from-equilibrium isons lead us to conclude that aHydro reproduces with high initial conditions might explain the unreasonable success of numerical accuracy the universal asymptotic attractor hydrodynamics in small systems such as pp and heavy-light obtained from the exact kinetic Gubser solution. Finally ion collisions [38]. we show that the asymptotic series solution of DNMR and In this work we continue exploring the properties of IS diverges asymptotically and we briefly comment on how attractors for rapidly expanding systems within relativistic to cure this problem by using a resurgent trans-series. kinetic theory. Previous works have focused on fluids The paper is structured as follows. In Sec. II we review undergoing Bjorken flow [20,26,27,36,37] and nonhomo- the Gubser flow, its exact solution for the Boltzmann geneous expanding plasmas [39]. We expand these studies equation within the relaxation time approximation (RTA) by investigating the properties of the attractors in the as well as different hydrodynamical truncation schemes. In plasmas undergoing Gubser flow [40,41]. Sec. III we study extensively the flow lines of the IS theory The Gubser flow describes a conformal system which in two and three dimensions. Our numerical studies of the expands azimuthally symmetrically in the transverse plane attractors for different theories and their comparisons are together with boost-invariant longitudinal expansion. The discussed in Sec. IV. In this section we also address the symmetry of the Gubser flow becomes manifest in the de issue of the divergences of the IS and DNMR hydrody- Sitter space times a line dS3 ⊗ R [40,41] and thus, the namical theories and how to fix them using resurgent dynamics of this system is studied in this curved spacetime. asymptotics. Our findings are summarized in Sec. V. Some The search for attractors in the Gubser flow poses new technical details of our calculations are presented in the challenges due to the geometry and the symmetries asso- Appendices. ciated with this velocity profile. We determine the location of the attractors with the help of well-known methods of II. SETUP nonlinear dynamical systems [42,43]. Our results bring new Before starting our discussion we first introduce the features and tools to the study of attractors which were not notation used in this paper. We work in natural units where addressed previously in the context of relativistic hydro- ℏ ¼ c ¼ kB ¼ 1. The metric signature is taken to be dynamics. For instance, in the two-dimensional (2D) “mostly plus” ð−; þ; þ; þÞ. In Minkowski space with system of ordinary differential equations (ODEs) derived from different hydrodynamical truncation schemes, we 1 study the flow diagrams—streamline plots of the velocity Previous studies of the Bjorken model have shown that after — rewriting the hydrodynamical equations for the temperature and vector fields in the space of state variables and carefully the only independent component of the shear viscous tensor in examine the early- and late-time behavior of the flow terms of the variable w ¼ τTðτÞ (where τ and T are the lines near each fixed point and show that the system is longitudinal proper time and temperature) reduce effectively to exponentially asymptotically stable. We observe that the a truly 1D nonlinear differential equation [20,26,27,36,37]. The attractor is a one-dimensional (1D) nonplanar manifold solution of this equation for a suitable initial condition determines what has been, by abuse of terminology, called the “attractor only asymptotically because the 2D system cannot be solution.” As we shall show in this work, this is not the case for dimensionally reduced to a 1D nonautonomous one by the Gubser flow.

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μ τ ϕ ς 3 ⊗ 1 1 ⊗ Milne coordinates x ¼ð ;r; ; Þ the line element is SOð Þq SOð ; Þ Z2 symmetry group: the de Sitter 2 given by time ρ, the combination of momentum components pˆΩ ¼ ˆ2 ˆ2 2 ˆ 2 μ ν 2 2 2 2 2 2 pθ þ pϕ/sin θ where pθ and the longitudinal momentum ds ¼ gμνdx dx ¼ −dτ þ dr þ r dϕ þ τ dς ; ð1Þ component pˆ ϕ are conjugate to the coordinates θ; ϕ, ˆ where the longitudinal proper time τ, spacetime rapidity ς respectively, as well as pς, which is conjugate to the ς 3 ⊗ and polar coordinates r and ϕ are related to the usual coordinate [9,13]. Thus, in dS3 R the RTA Boltzmann Cartesian coordinates (t, x, y, z) through the following equation reduces to a one-dimensional relaxation-type expressions: equation [9,13]      pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 −uˆ · pˆ 2 2 z ∂ρfðρ; pˆ ; pˆςÞ¼− fðρ; pˆ ; pˆςÞ − f ; τ ¼ t − z ; ς ¼ arctanh ; Ω τˆ ρ Ω eq ˆ ρ t rð Þ Tð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   y ð5Þ r ¼ x2 þ y2; ϕ ¼ arctan : ð2Þ x ˆ where T is the temperature. In this work we use ∂ρ to mean ρ −z It is better to study the Gubser flow in de Sitter space times d/d and take feqðzÞ¼e as the local thermal equilibrium ⊗ a line (dS3 R) that is a curved spacetime in which the distribution. The conformal symmetry demands τˆrðρÞ¼ flow is static and the symmetries are manifest [40,41]. c/TˆðρÞ with c ¼ 5η/s where η and s are the shear viscosity This is obtained by applying a conformal map between and entropy density, respectively. Given a solution for dS3 ⊗ R and Minkowski space, which consists of the Boltzmann equation (5) one calculates the energy- 2 2 Ω 2 −2 rescaling the metric ds →dsˆ ¼e ds with Ω ¼ logτ . momentum tensor as a second-rank tensor moment Afterwards one performs the coordinate transformation defined as4 xμ ¼ðτ;r;ϕ; ηÞ ↦ xˆμ ¼ðρ; θ; ϕ; ηÞ2 with   Tˆ μνðρÞ¼hpˆμpˆνi: ð6Þ 1 − τ˜2 ˜2 ρ τ˜ ˜ − þ r ð ; rÞ¼ arcsinh 2τ˜ ; ˆ μν   The relaxation equations of T are obtained by either 2˜ solving Eq. (5) exactly or by finding an approximate θ τ˜ ˜ r ð ; rÞ¼arctan 2 2 : ð3Þ perturbative solution. In the next section we review differ- 1 þ τ˜ − r˜ ent approximate methods to obtain the fluid dynamical Here, τ˜ ¼ qτ and r˜ ¼ qr with q being an arbitrary energy equations within kinetic theory as well as the exact solution scale that sets the transverse size of the system [40,41]. The of the kinetic Eq. (5). time-like coordinate is the variable ρ ∈ ð−∞; ∞Þ and the polar coordinate is θ ∈ ½0; 2πÞ. Therefore, the line element B. Fluid dynamical theories ⊗ in dS3 R reads as For weakly coupled systems the equations of the macro- scopic fluid dynamical variables are derived from a micro- ˆ2 − ρ2 2ρ θ2 2θ ϕ2 ς2 ds ¼ d þ cosh ðd þ sin d Þþd ; ð4Þ scopic underlying kinetic theory based on the Boltzmann equation. The derivation of these equations assumes that ˆ 2 2 2 so the metric is gμν ¼ diagð−1; cosh ρ; cosh ρsin θ; 1Þ.In the Boltzmann equation admits a generic solution of the dS3 ⊗ R the flow velocity is the normalized time-like form ˆ μ 1 0 0 0 vector u ¼ð ; ; ; Þ, which is invariant under the X q-deformed symmetry group SOð3Þ ⊗ SOð1; 1Þ ⊗ Z2 μ μ μ ðlÞ μ q fðx ;piÞ¼fbðx ;piÞ aαðx ÞMα ðx ;piÞ; ð7Þ [40,41]. We choose the fluid velocity to be defined in α;l ˆ μν μ the Landau frame, i.e., T uˆ ν ≡ ϵˆuˆ . where fb is a distribution function that describes the ðlÞ A. Relativistic kinetic theory for the Gubser flow evolution of an existing background, Mα are orthogonal polynomials of degree l, whose orthogonality properties In kinetic theory the macroscopic hydrodynamic varia- depend on f0, and aα are moments of the full distribution bles are calculated as momentum moments of the distri- function f. The leading-order background distribution bution function. The symmetry of the Gubser flow restricts 2 function is chosen based on the problem at hand [62]. the phase-space distribution fðx;ˆ pˆÞ¼fðρ; pˆΩ; pˆςÞ, i.e., the distribution function depends only on invariants of the 3A more detailed derivation of this exact solution can be found in the original references [9,13]. 2 4 ˆ μ We assign variables with a hat to all quantities defined in Any phase-space observable Oðx;ˆ pˆ Þ obtained from a given ⊗ O ˆ ˆ dS3 R. phase-space distribution fX will be denoted as h ðx; pÞiX.

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In the rest of the section we shall briefly describe two energy density. As a consequence, deviations from the local choices of the leading-order background distribution func- thermal equilibrium are captured entirely by the shear stress μν hμ νi tion which leads to different relaxation equations for the tensor πˆ ≡ hpˆ pˆ iδ where h iδ indicates a momentum components of the energy-momentum tensor. moment weighted by δf. ˆ ˆ μν The energy-momentum conservation law DμT ¼ 0 for – 1. Expansion around an equilibrium background the energy-momentum tensor (10a) (10c) gives us the following evolution equation for the energy density ϵˆ The canonical derivation of fluid dynamics assumes an [40,41]: expansion around a local equilibrium background   ∂ρϵˆ 8 πˆ ˆ ˆ þ tanh ρ ¼ tanh ρ; ð11Þ ˆ ˆ − u · p δ ˆ ˆ ϵˆ 3 ϵˆ fðx; pÞ¼feq þ fðx; pÞ: ð8Þ TˆðρÞ which can be rewritten in terms of the temperature since for ˆ 4 ˆ ˆ For relativistic systems feq is taken to be the equilibrium conformal systems ϵˆ ∼ T and thus, ∂ρT/T ¼ 4∂ρϵˆ/ϵˆ.Asa μ β½ðu·pÞ−μ Jüttner distribution function fðx ;piÞ¼1/ðe þ aÞ result one finds [40,41] where β ¼ 1/T with T being the temperature, μ the ˆ chemical potential and a ¼þ1; −1, 0 for particles follow- ∂ρT 2 π¯ þ tanh ρ ¼ tanh ρ; ð12Þ ing Fermi-Dirac, Bose-Einstein or Maxwell-Boltzmann Tˆ 3 3 statistics, respectively. The energy of the particle −uˆ · pˆ is taken to be isotropic in the local rest frame (LRF) while where π¯ ≔ πˆςς/ðϵˆ þ Pˆ Þ. We point out that for the Gubser δf encodes the deviations from the thermal equilibrium flow there is only one independent component of the shear δ ≪ state. It is implicitly assumed that f feq. For systems viscous tensor [40,41]. The additional evolution equation close to equilibrium the four-momentum is decomposed needed for π¯ is obtained by expanding δf within some μ ˆ μ hμi ˆ δ as pˆ ¼ Epˆ uˆ uˆ þ pˆ where Epˆ uˆ ¼ −ðuˆ · pˆÞ (in the LRF approximation. The form of f is found by expanding it in ˆ ρ hμi ˆ μν ˆ Epˆ uˆ ≡ pˆ ) while pˆ ¼ Δ pˆν projects over the spatial terms of a set of orthogonal polynomials in energy Epˆ uˆ, and momentum component orthogonal to the flow velocity. For a set of irreducible tensors in momentum invariant under the Gubser flow this vectorial decomposition of the four- the little group SOð3Þ of the Lorentz transformations, i.e., momentum allows to write the most general conformal 1, phμi, phμpνi, etc. Afterwards, the relaxation equations of isotropic energy-momentum tensor [63] the dissipative macroscopic quantities are obtained by applying a systematic truncation method based on a power μν Tˆ μν ¼ ϵˆuˆ μuˆ μ þ Pˆ Δˆ þ πˆμν; ð9Þ counting in the Knudsen Kn and inverse Reynolds Re−1 numbers [45]. The lowest truncation order provides the IS where ϵˆ is the energy density, Pˆ is the isotropic pressure equations while the inclusion of all other second-order and πˆμν is the shear stress tensor, which are the momentum terms, i.e. in Kn2, Re−2 and Kn · Re−1, gives the DNMR moments of the distribution function given by equations. A detailed discussion of these power-counting schemes can be found in Ref. [45]. 2 ˆ ϵˆ ¼hð−uˆ · pˆÞ i; ð10aÞ For the normalized shear stress π¯ ¼ πˆ/ðϵˆ þ PÞ one finds the following evolution equations for the IS and DNMR 1 ˆ ˆ μ ν theories, respectively [13,64]: P ¼ hΔμνpˆ pˆ i; ð10bÞ 3   dπ¯ 4 2 4 η μν μ ν τˆπˆ þ ðπ¯Þ tanh ρ þ π¯ ¼ tanh ρ; ð13aÞ πˆ ¼hpˆh pˆ ii: ð10cÞ dρ 3 3 sTˆ   In Eq. (10c) we introduce the symmetric, orthogonal to uˆ μ 4 4 η τˆ ∂ π¯ π¯ 2 ρ π¯ ρ hμ νi Δˆ μν α β Δˆ μν πˆ ρ þ ð Þ tanh þ ¼ ˆ tanh and traceless operator A B ¼ αβA B with αβ ¼ 3 3 sT ˆ μ ˆ ν ˆ μ ˆ ν 2 ˆ μν ˆ ðΔαΔβ þ ΔβΔα − 3 Δ ΔαβÞ/2. For the Gubser flow there 10 ςς þ τˆπˆπ¯ tanh ρ: ð13bÞ is only one independent shear stress component πˆ ≡ πˆ 7 [40,41]. For conformal systems the equation of state reads τˆ ˆ 5η as Pˆ ¼ ϵˆ/3. The conformal symmetry implies πˆ ¼ c/T with c ¼ /s. The local temperature of the system introduced in Eq. (8) ϵˆ ϵˆ C. Anisotropic hydrodynamics is found from the Landau matching condition ¼ eqðTÞ [1]. This phenomenological constraint ensures that the Anisotropic hydrodynamics [46–61] attempts to describe ˆ parameter T in feq is adjusted such that the dissipative systems with a large expansion rate along some particular corrections encoded in δf do not shift the value of the direction lˆμ. In these situations there is a momentum

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ˆμ P anisotropy along l driving the system far from equilib- 1. L matching rium. Thus, instead of expanding around an equilibrium When introducing the leading-order anisotropic distri- configuration, it is better to consider an anisotropic back- bution function defined in Eq. (14) we did not comment on ground as a leading-order term for the generic expansion of how to determine the parameters Λˆ and ξ. The different the Boltzmann equation (7), i.e., variants of aHydro have been proposed in recent years in –   order to perform this matching procedure [46 61].Ina uˆ · pˆ ξ previous study of the Gubser flow, it was shown that the Pˆ fðx;ˆ pˆÞ¼f − ; ðlˆ · pˆÞ þ δf˜ðx;ˆ pˆÞ: ð14Þ L a Λˆ Λˆ matching scheme [48] provides the most accurate macro- scopic description when comparing its predictions with The parameter ξ measures the strength of momentum the ones obtained from an exact solution of the RTA μ Boltzmann equation. Thus, we shall consider in this work anisotropy along the lˆ direction, Λˆ is an arbitrary this matching procedure. δ ˜ momentum scale and f takes into account residual We start by introducing first the anisotropic integrals dissipative corrections. In principle, the parameter ξ mea- n−l−2q ˆ l ˆ μ ν q sures the size of the microscopic momentum-space anisot- Inlq ¼hð−uˆ · pˆÞ ðl · pˆÞ ðΞμνpˆ pˆ Þ i ropies. For anisotropic systems the four-momentum of ≡ ˆ Λˆ ξ ˜ ˆμ ˆ ˆ μ ˆ ˆμ ˆfμg Inlqð ; ÞþInlq: ð18Þ particles is decomposed as p ¼ Epˆ uˆ u þ Epˆ lˆl þ p ˆ ˆ ˆ where Epˆ lˆ ¼ðl · pÞ is the spatial component of the particle The first term on the rhs of the previous equation comes ˆμ ˆμ ˆ from the leading-order contribution associated with the momentum along the space-like vector l (l lμ ¼ 1) while fμg ˆ μν anisotropic distribution function fa [Eq. (14)] while the pˆ Ξ pˆν ¼ are the spatial components of the momentum second term corresponds to the subleading contribution ˆ μ ˆμ which are orthogonal to both u and l . Here, we introduce from δf˜ in Eq. (14). In Appendix A we show how to Ξˆ μν ˆμν ˆ μ ˆ ν − ˆμ ˆν Δˆ μν − ˆμ ˆν the projection tensor ¼ g þ u u l l ¼ l l perform the integrals of the leading-order anisotropic μ ˆμ which is orthogonal to uˆ and l [48,54,55,65–68]. We also distribution function for the massless case. The functional fμ νg ˆ μν α β define the symmetric traceless operator A B ¼ ΞαβA B form of fa is taken to be the RS ansatz which in the LRF ˆ μν μ ν ν μ ˆ μν looks like [69] with Ξαβ ¼ðΞαΞβ þ ΞβΞα − Ξ ΞαβÞ/2. By construction Ξˆ μν ˆ μ ˆμ αβ is also orthogonal to u and l . ˆ ˆ; Λˆ ξ ξ Λˆ faðx; p ; Þ¼feqðERSð Þ/ Þð19Þ The anisotropic decomposition for pˆ μ allows us to write the most general anisotropic energy-momentum tensor as −z where feqðzÞ¼e is a Maxwellian distribution function [48,55] evaluated for the momentum-anisotropic argument qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ μν ϵˆˆ μ ˆ ν Pˆ ˆμ ˆν Pˆ Ξˆ μν T ¼ u u þ Ll l þ ⊥ ; ð15Þ ξ ≡ ˆ ˆ 2 ξ ˆ ˆ 2 ERSð Þ ðu · pÞ þ ðl · pÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where the energy density ϵˆ, transverse and longitudinal ¼ pˆΩ/ðcosh ρÞþð1 þ ξÞpˆς: ð20Þ ˆ ˆ pressures P⊥ and PL, respectively, are the following momentum moments: The leading-order anisotropic variables contributing to the anisotropic energy-momentum tensor (15) are ϵˆ ˆ ˆ ˆ μν ≡ −ˆ ˆ 2 ¼ uμuνT hð u · pÞ i; ð16aÞ ˆ μν ϵˆ ˆ μ ˆ ν Pˆ RS ˆμ ˆν Pˆ RSΞˆ μν TRS ¼ RSu u þ L l l þ ⊥ ; ð21Þ

1 1 where Pˆ Ξˆ ˆ μν ≡ Ξˆ ˆμ ˆν ⊥ ¼ 2 μνT 2 h μνp p i; ð16bÞ ϵˆ −ˆ ˆ 2 ˆ Λˆ ξ RS ¼hð u · pÞ ia ¼ I200ð ; Þ; ð22aÞ μν ˆ ˆ ˆ ˆ ˆ 2 2 PL ¼ lμlνT ≡ hðl · pˆÞ i: ð16cÞ Pˆ RS ˆ ˆ ˆ Λˆ ξ L ¼hðl · pÞ ia ¼ I220ð ; Þ; ð22bÞ ϵˆ 1 1 In this case the conformal symmetry implies that ¼ Pˆ RS Ξˆ ˆμ ˆν ˆ Λˆ ξ ˆ ˆ ⊥ ¼ 2 h μνp p ia ¼ 2 I201ð ; Þ: ð22cÞ 2P⊥ þ PL. For the Gubser flow the shear stress tensor can be related to the total pressure anisotropy through [48,55] The traceless condition of the conformal anisotropic energy-   ϵˆ 2Pˆ RS Pˆ RS 2 1 momentum tensor implies that RS ¼ ⊥ þ L .The μν ˆ ˆ ˆμ ˆν ˆ μν πˆ ¼ ðPL − P⊥Þ l l − Ξ : ð17Þ conservation law gives the evolution equation for the energy 3 2 density

044041-5 BEHTASH, CRUZ-CAMACHO, and MARTINEZ PHYS. REV. D 97, 044041 (2018)   8 2 π¯ 4 5 9 ∂ ϵˆ ϵˆ ρ Pˆ RS − Pˆ RS ρ ∂ π¯ ρ π¯ − π¯2 − F π¯ ρ RS þ 3 RS tanh ¼ 3 ð L ⊥ Þ tanh : ð23Þ ρ þ ¼ tanh þ ð Þ ; ð28Þ τˆr 3 16 16

The matching condition for the energy density where ϵˆ Λˆ ξ ϵˆ RSð ; Þ¼ eqðTÞ leads to the following relation between ˆ ˆ Rˆ ξ π¯ the temperature T and the momentum scale Λ: F π¯ ≡ 240ð ð ÞÞ ð Þ ˆ : ð29Þ R200ðξðπ¯ÞÞ Tˆ Λˆ ¼ : ð24Þ ˆ 1/4 Rˆ Rˆ ðR200ðξÞÞ The functions 240 and 200 are listed in Appendix A. Furthermore, ξðπ¯Þ in Eq. (29) is the inverse of the function ˆ The function R200ðξÞ is given in Eq. (A3). This relation ensures that the energy density does not receive any πˆ 3 ˆ − ˆ π¯ ξ I220 I200 δ ˜ ð Þ¼ ˆ ¼ ˆ contribution from the residual deviation f in Eq. (14). ϵˆ þ P 4I200 Now, for the Gubser flow the conservation law (23)   1 3Rˆ ξ indicates that the pressure anisotropy is the force that 220ð Þ − 1 ¼ 4 ˆ : ð30Þ drives the system far from equilibrium. The microscopic R200ðξÞ origin of this force is the momentum anisotropy created by the expansion of the system which in the LRF is measured An advantage of using aHydro is that by construction the by the anisotropy parameter ξ. Hence, one simply adjusts transverse and longitudinal pressures remain positive dur- ξ the value of such that the leading-order anisotropic ing the entire evolution of the system as is expected in distribution function fa fully captures the information of kinetic theory.6 These constraints imply −1/4 < π¯ < 1/2 the full pressure anisotropy. For the Gubser flow this means which is in agreement with the anisotropy parameter Pˆ that the longitudinal pressure L [Eq. (16c)] does not ξ ∈ ð−1; ∞Þ. On the other hand, the lhs of Eq. (30) is δ ˜ receive contributions from the residual deviation f of the identified as a termpffiffiffiffiffiffiffiffiffiffiffiffiffi proportional to the inverse Reynolds Pˆ Pˆ RS −1 π πμν P P ϵˆ distribution function (14), i.e. L ¼ L . Now the effective number Re ¼ μν / 0 (with 0 ¼ /3 for the con- shear viscous tensor is related to the total pressure formal case) [45]. Equation (30) therefore indicates that the anisotropy via the identity (17) which in this case gives [48] anisotropy parameter resums nonperturbatively not only −1 2 large gradients (i.e., Knudsen number) but also large Re πˆ Pˆ RS − Pˆ RS numbers. An analogous relation between the effective shear RS ¼ 3 ð L ⊥ Þ and the anisotropy parameter was found in the Bjorken case 1 ˆ ˆ 2 − −ˆ ˆ 2 [46]. The dissipative corrections OðRe−2Þ arise in general ¼hðl · pÞ ð u · pÞ ia 3 from the most nonlinear sector of the collisional kernel [45] ˆ ˆ 1 ˆ ˆ and thus, the calculation of these terms is cumbersome even ¼ I220ðΛ; ξÞ − I200ðΛ; ξÞ: ð25Þ 3 for the simplest kernels [70]. Moreover, some of the nonlinear terms OðRe−2Þ calculated in the DNMR theory Pˆ Thus, the L matching prescription can be recast into lead to violations of causality [71]. the following condition for the effective shear viscous component: D. Exact solution to the Boltzmann equation in the RTA approximation πˆ πˆ ¼ RS: ð26Þ The RTA Boltzmann equation (5) admits the following Furthermore, Eq. (25) allows us to rewrite the conservation exact solution [9,13]: law (23) as Z 1 ρ 2 2 0 fexðρ; pˆ Ω; pˆ ςÞ¼Dðρ; ρ0Þf0ðρ0; pˆ Ω; pˆ ςÞþ dρ 8 c ρ0 ∂ρϵˆRS þ ϵˆRS tanh ρ ¼ πˆRS tanh ρ: ð27Þ 3 0 ˆ 0 ˆ 0 ˆ 0 × Dðρ; ρ ÞTðρ ÞfeqðEpˆ ðρ Þ/Tðρ ÞÞ; ð31Þ Pˆ πˆ For the L matching, the equation for RS is found from Eq. (25). After some algebra one finds the following 6The positive-definite condition of the longitudinal and trans- evolution equation for the normalized effective shear π¯5: verse pressures is not a requirement for holographic models within the AdS/CFT correspondence since the quasiparticle picture does not exist in these theories. In weakly coupled 5Technical details of this derivation are discussed in Sec. III C theories where the quasiparticle picture exists, the positivity of of Ref. [48]. the pressure is violated in the presence of external fields.

044041-6 FAR-FROM-EQUILIBRIUM ATTRACTORS AND NONLINEAR … PHYS. REV. D 97, 044041 (2018) R 1 ρ 0 ˆ 0 D ρ; ρ0 − dρ T ρ where ð Þ¼exp ½ c ρ0 ð Þ is the damping and at late times they show what happens to the matter function. For the initial condition of the distribution distribution until it evolves to a steady state (thermally 7 function f0 at ρ0 we shall consider the RS ansatz [69] nonequilibrium) at ρ → ∞. As a minor digression, we recall that for the Bjorken  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ˆ 2 flow, the second-order hydrodynamical equations can be ρ ; ˆ 2 ˆ − pΩ 1 ξ ˆ 2 f0ð 0 pΩ; pςÞ¼exp ˆ 2 þð þ 0Þpς ; entirely reduced to one single explicitly time-dependent Λ cosh ρ0 0 ODE (i.e. 1D nonautonomous parametrized by a new time ð32Þ w) due to scaling symmetry [19,25]. This suggests that its attractor is a planar 1-manifold characterized by w and the ˆ where Λ0 is the initial temperature and ξ0 is the initial basin of attraction is indeed 2D. But what is the proper momentum anisotropy along the ς direction. From the exact definition of an attractor? It is an invariant set of points and solution for f one gets the energy density and the only every flow line starting from a point within the field of independent component of the shear stress [9,13]: attraction of the attracting fixed point, will always limit to this set of points at late times. Given a set of initial values ˆ ϵˆðρÞ¼I200; for Bjorken time, the flow lines always lie on a plane and as   4 long as they belong to a special set of numbers, they tend to cosh ρ0 ˆ ¼ Dðρ; ρ0Þ ϵˆ ðΛ0; ξ ðρ; ρ0; ξ0ÞÞ the attractor at late times. This set defines the basin of cosh ρ RS FS Z   attraction: given an initial time τ0, there is an associated ρ 0 4 ˆ 1 cosh ρ Tðτ0Þ and π¯ðτ0Þ, and the basin of attraction is elaborated as dρ0D ρ; ρ0 Tˆ ρ0 þ ð Þ ð Þ ρ ˆ τ π¯ τ c ρ0 cosh the set of all pairs of ðTð 0Þ; ð 0ÞÞ such that the Bjorken ϵˆ ˆ ρ0 ξ ρ; ρ0 0 flow lines are doomed to limit to the equilibrium. We recall × RSðTð Þ; FSð ; ÞÞ; ð33aÞ that ðT;ˆ π¯Þ is the actual phase space8 of the Bjorken flow 1 which fixes the dimensionality of the basin of attraction πˆ ρ ˆ − ˆ ð Þ¼I220 3 I200; as well.   4 In the case of Gubser flow, we simply identify the fixed cosh ρ0 ˆ ¼ Dðρ; ρ0Þ πˆ ðΛ0; ξ ðρ; ρ0; ξ0ÞÞ points and discuss whether or not they are stable by cosh ρ RS FS Z   analyzing the eigenvalues of the Jacobian matrix of the 1 ρ cosh ρ0 4 linearized flow equations near each fixed point. We find ρ0 ρ ρ0 ˆ ρ0 þ d Dð ; ÞTð Þ ρ that there are two unstable saddle points and one stable c ρ0 cosh fixed point which determines the steady state of the system πˆ ˆ ρ0 ξ ρ; ρ0 0 × RSðTð Þ; FSð ; ÞÞ: ð33bÞ in any hydrodynamics theory. This task is carried out by analyzing the null lines of the 2D nonautonomous system ξ ρ; ρ ξ −1 1 ξ cosh ρα 2 We note that FSð α; αÞ¼ þð þ αÞð cosh ρ Þ . (34a)–(34b). The intersection of these lines gives the fixed These integral equations are solved numerically by means points. By analyzing the asymptotics of flow lines close to of the method described in Refs. [11–13,72]. The temper- the steady state, it is determined that the fixed point is ature of the system is obtained from the energy density indeed exponentially asymptotically stable. One observa- ϵˆ Λˆ ξ through the Landau matching condition, i.e., RSð ; Þ¼ tion that we make is that the flow equations cannot be 3 π2 Λˆ 4Rˆ ξ ˆ 4 π2 reduced to one single nonautonomous ODE by any kind of ð / Þ 200ð Þ¼3T / . reparametrization unless at very late times such that ρ is 2 III. FLOW DIAGRAMS OF THE IS THEORY large enough that we can take tanh ρ ∼ 1. But this is fine from the viewpoint of attractors since they are asymptotic In dS3 ⊗ R the expansion rate of the Gubser flow or in other words limit sets. This does not affect the ˆ θ ¼ 2 tanh ρ becomes a constant when ρ → ∞.Aswe dimensionality of the attractor and hence Gubser flow lines shall see below this geometrical property of the velocity still evolve into a 1-manifold but in reality it is no longer a profile poses a challenge to find the attractors of different planar curve since the time direction is not fully decoupled hydrodynamical theories. In this section we explain a from the ODE in terms of the new time w, which will be method for finding the attractors based on the mathematical theory of nonautonomous systems [42,43]. We discuss 7Throughout this paper the word “equilibrium” is always extensively the case of IS theory. The same method can be meant to be thermal equilibrium and a steady state expresses a used in different models. state of the system which requires energy or continual work to The gist of what we are about to do in this section is to remain stationary and it thus does not imply thermal equilibrium. 8 “ ” study the flow diagrams of the Gubser flow from the In this section, phase space is referred to as the space of independent macroscopic variables labeling a flow diagram. The perspective of nonlinear dynamical systems. The flow lines latter represents the dynamics of the underlying hydrodynamical at early times do dictate the far-from-equilibrium behavior system via the connection between the velocity fields and the of any system governed by a set of differential equations macroscopic variables that form the flow lines.

044041-7 BEHTASH, CRUZ-CAMACHO, and MARTINEZ PHYS. REV. D 97, 044041 (2018) clarified in Sec. III B 1. What about the basin of attraction? These two equations can be combined in such a way that We study this point in great detail in Sec. III B 2 where we we are left with the following second-order nonlinear observe that the impossibility of this dimensional reduction nonautonomous differential equation for the temperature to a 1D system is a symptom of the peculiarity of the TˆðτÞ in the IS theory: 3 q-deformed conformal symmetry SOð Þq which is not a simple scaling of variables and time as in the Bjorken case. Tˆ 2ðτÞ 135cτTˆ 02ðτÞþ ð76cτ3 − 45τðτ2 − 1ÞTˆ 0ðτÞÞ The variable τ ¼ tanh ρ is promoted to be independent ðτ2 − 1Þ2 which lifts the phase space dimensionality by 1, now being 2 ˆ 3 ˆ ˆ 30τ T ðτÞ 15cTðτÞ labeled by ðT;π¯; τÞ. Therefore, we lift the dynamical þ − ðτ2 − 1Þ2 ðτ2 − 1Þ system describing the flow equations to a 3D autonomous   system where ρ dependency is now implicit. So the basin of 3τ 13τ2 − 3 ˆ 0 τ − ˆ 00 τ 0 attraction is three dimensional, being identical to the × ð ÞT ð Þ τ2 − 1 T ð Þ ¼ : dimensions of the autonomous system (or phase space). We finally emphasize the importance of the basin of To find the fixed points of the system, we solve ˆ attraction and the way to define it locally and globally. As Eqs. (34a)–(34b) for π¯N and TN along the respective null was said before, the basin of attraction defines an effective lines. (An A null line is a trajectory in the phase space along attractive potential field for the flow lines, thus forcing which dA/ds ¼ 0 where s is the flow time variable.) We them to evolve into the stable steady state at late times. This first consider the case where the Tˆ null line is given by the is an important problem in far-from-equilibrium hydro- solution π¯NðτÞ¼2 to Eq. (34a) with vanishing temperature dynamics, i.e., to map out all the divergent flows and only derivative, and TˆðτÞ is a nonzero constant. The Gubser π¯ probe the relevant stable ones. Similar to the path integral null line equation, where a steepest descent path is one that counts as contributing to the path-integral manifold, in hydrodynam- 4 2 1 4 π¯ ðτÞτ þ π¯ðτÞTˆ ðτÞ − τ ¼ 0 ð35Þ ics the space of all flow lines taking initial values in the 3 N c N 15 basin of attraction would determine this manifold. ˆ τ −38 τ Effectively, it is just enough to consider all the paths then fixes the value of the temperature at TNð Þ¼ c /15, τ ∼ 1 ˆ −38 starting at ρ → −∞ somewhere on the boundary of the which at large time is simply Tc ¼ c/15. The basin where the attractive force field of the fixed point is the intersections of these null lines are not generally reached by τ ∼ 1 weakest. This field is determined by an effective potential all flow lines since near there are flow lines, for function(al) known as the Lyapunov function V that instance, that either diverge from or converge to this point. satisfies two key properties: V > 0, and dV/dρ ≤ 0. The Furthermore, knowing that the system converges to a fixed τ ≫ 0 existence of V hints at the stability of the steady state and point sometime at late times , it is certainly not physical that the linearized flow equations are asymptotically stable. to consider that the point to be reached has a negative V temperature ð−38c/15Þ and thus we are led to a situation The local function loc can always be computed in the near- equilibrium region by solving the Lyapunov equation given where we look for the stable steady state in the range π¯ π¯ 2 in Sec. III B 2 and we hence solve it to estimate the shape of ¼ c < .Figure1 indicates explicitly why this point the local basin of attraction. The global function is difficult cannot define a stable steady state for the system. Taking this to precisely build but there are numerical and analytical to be the case, one may solve for temperature from Eq. (34a), optimization techniques that are mentioned in the same to only obtain subsection. The global Lyapunov function mimics the 2 2−π¯c ˆ τ ˆ 1 − τ 6 behavior of all stable fluctuations both at early and late Tð Þ¼T0ð Þ ; ð36Þ times which in this regard might be a very useful tool to ˆ for a constant T0 > 0. Note that this represents the evolution construct an effective partition function for hydrodynamics. of temperature near τ ¼ 1 and it is always positive for all We will close this section by commenting on this issue. times. We can check that A. From the perspective of a 2D nonautonomous dTˆ τ dynamical system lim ð1 − τ2Þ ¼ − lim ð2 − π¯ ÞTˆðτÞ¼0: ð37Þ τ→1− dτ τ→1− 6 c Using a secondary time parameter τ ¼ tanh ρ ∈ ½−1; 1, Eqs. (12) and (13b) are put into the forms Inserting this temperature into Eq. (34b) gives the shear along the π¯ null line toward the stable steady state, dTˆ τTˆ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! π¯ τ − 2 2 2 ¼ 2 ð ð Þ Þ; ð34aÞ 3 64 τ 2−π¯ 2−π¯ τ 3 1 − τ c 2 c 2 c d ð Þ π¯ðτÞ¼ þð1 − τ Þ 3 − ð1 − τ Þ 6 :   N 8cτ 45 dπ¯ 1 4 1 4 ¼ − π¯2ðτÞτ þ π¯ðτÞTˆðτÞ − τ : ð34bÞ dτ 1 − τ2 3 c 15 ð38Þ

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(a) 1 2 ffiffiffi π¯ ðτÞ¼∓ p ffiffiffi 8 4 ; ð39Þ p pffiffi 2 pffiffi 5 5ðe3 5ð1 − τ Þ3 5 þ 1Þ

which demonstrates the evolution of the shear component of the trajectories near the fixed points ð0; πc Þ. The fixed points in the IS theory of the Gubser flow are therefore determined to be 1 Fixed points∶ π¯ ¼pffiffiffi ; Tˆ ¼ 0; and c 5 c 38 π¯ 2 ˆ − c c ¼ ; Tc ¼ 15 ; ð40Þ which is consistent with the flow diagrams plotted in Fig. 2. We can also read the Lyapunov exponent9 of Tˆ from the (b) asymptotic solution (36), for both fixed points with pffiffiffi π¯ 1 ˆ ρ ∼ c ¼ / 5, about which temperatureffiffiffi goes like T ð Þ λρ p ˆ ˆ 1 T0e T where λ − 2 ∓ 1/ 5 , thus hinting at the Tˆ ¼ 3 ð Þ exponentially fast running up of the flow lines at ρ → ∞ to the stable steady state and even faster convergence to the repelling point only along the attracting direction. Along the pffiffiffi shear component of the flow trajectories close to 0; 1/ 5 pðffiffiffi Þ the Lyapunov exponent is given by λπ¯ ¼ −8/ð3 5Þ, which is λþρ a sign of seemingly drastic convergence ð∼e π¯ Þ to the steady state or divergence from the repeller. This provides de facto evidence of the asymptotically exponential stability of the steady state in the Gubser model which can be rigorously derived by building Lyapunov functions of the linearized system. FIG. 1. IS flow lines of the Gubser model for (a) early time and We will be mainly interested in the late-time behavior of (b) late time. The black dots indicate the phase-spacepffiffiffi fixed points. the Gubser flow near the attracting fixed point. We refer to 0 −1 In panel (a) the repelling fixed point ð ; / 5Þ at early times this as an “attractor” that points to the existence of some (here τ −0.9) directs the flows in the basin towards the attractor ¼ pffiffiffi BA 3A 2 −1 A bounded set, e , namely an absorbing set, in the phase at late times. Notice that s þ ¼ / 5 where s defines the X ˆ π¯ initial condition in Eq. (43). Physically, this is the situation where space of all independent state variables ðT; Þ and τ BA the distribution of matter is most anisotropic, and thus farthest possibly time : e is indeed invariant under the forward from the stable steady state. In panel (b) the flow diagram shows evolution of state equations in time such that every solution A that the stable fixed point (40) is approached by the flow lines at of the system of ODEs has to ultimately enter Be provided late times (here τ ¼ 0.9). Note that, in terms of the new variable τ that the chosen initial conditions for the flow trajectories the time flows from τ ¼ −1 to τ ¼ 1 and therefore an observer put them in a bounded set Be known as the “basin of with a line of sight along the time direction would see the flow pffiffiffi attraction” for the attracting fixed point. Notice that by way lines coming out of the point ð0; 1/ 5Þ at early times and going in A τ of definition, Be ⊂ Be. Let ϕ be the flow defined by the at late times, which suggests that the basin of attraction is three Gubser dynamical system, Eqs. (34a)–(34b); then we can dimensional. At late times the flow diagram for Gubser flow in the IS theory at ∼τ ¼ 0.9 shows that there is one attracting fixed define an attractor for this system by point e.g. a sink, and two repelling fixed points e.g. saddle points. The area trapped between the green lines on the top right 9The Gubser flow as a dynamical system is deterministic in the represents the physical portion of the basin of attraction at a sense that the stability of its fixed points is quantified by the given time slice. eigenvalues of the linearization matrix [the so-called Jacobian matrix, cf. Eq. (49)] at these points. Therefore, the Lyapunov exponents are the real parts of the eigenvalues of this matrix 2 dπ¯ computed along a flow trajectory that satisfies the ordinary We also notice that limτ→1− ð1 − τ Þ N vanishes as expected. dτ pffiffiffi differential Eq. (47). The more interesting case of deterministic − At the limit τ → 1 , this yields the value of π¯c ¼1/ 5. chaos occurs when the maximal Lyapunov exponent is positive. Similarly the ray Tˆ ¼ 0 is a Tˆ null line, along which Eq. (34a) In the current system under study, or in any known hydrody- pffiffiffi namical system, the maximal Lyapunov exponent is always can be solved subject to the conditions π¯ð∞Þ¼1/ 5 negative near a stable fixed point which does not give rise to to give chaos.

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this may not be the case unless it could be reduced to a single nonautonomous ODE, by a reparametrization of state variables and τ. In the case that such a reduction is possible, the attractor A can be formally defined using, instead of the flows ϕτ, the map AðwÞ∶U → R for some U ⊆ R ≔ ˆ τ ¼½wmin;wmax , where for instance, w wðT; Þ, and AðwÞ is an algebraic function of π¯ subject to the boundary conditions

A → A A → A ðw wmaxÞ¼ e and ðw wminÞ¼ s ð43Þ

ˆ A−1 τ ∈ BA ˆ A−1 τ ∈ ∂B 10 such that ðTe; e ; maxÞ e and ðTs; s ; minÞ e. ˆ A−1 τ We keep in mind that ðTs; s ; minÞ is basically a saddle point in second-order hydrodynamical theories in which the underlying dynamical system entails a term proportional to π¯2. Finally, the one-dimensional manifold of A can be represented by

A ∼ AðwÞ: ð44Þ w≫w FIG. 2. The late-time behavior of flow lines in thep ISffiffiffi theory of min Gubser flow. The wedge near the fixed point ð0; 1/ 5Þ asymp- totically evolves into a 1D manifold that represents the behavior Let us introduce the parametrization w ¼ tanh ρ/Tˆ of of a Gubser attractor at late times. The other two fixed points are Gubser time and the function AðwÞ defined as clearly saddle points of which the lower one in the plot feeds the attractor at early times. The flow line lying on the segment BA has ˆ ˆ 1 ∂ρT d logðTÞ the fastest convergence among all other flows due to the repelling AðwÞ¼ ¼ : ð45Þ nature of the saddle point. The coloring of the flow lines is tanh ρ Tˆ d logðcosh ρÞ implemented in such a way that one could perceive the depth of the 3D basin of attraction by looking into the flow diagram Using these definitions, the evolution equations of the IS perpendicularly. theory [Eqs. (12) and (13a)] boil down to the following ODE: τ A A ¼ ⋂ ϕ ðBe Þ; ð41Þ τ≥−1 dAðwÞ 4 3wðcoth2ρ − 1 − AðwÞÞ þ ð3AðwÞþ2Þ2 dw 3 which will be compact and invariant, and subject to the 3AðwÞþ2 4 condition þ − ¼ 0; ð46Þ cw 15 ϕ−1 ∉ X B n e ð42Þ where π¯ ¼ 3AðwÞþ2 from the conservation law (12). This reduces the dimensionality of the problem only every flow trajectory will approach this set as τ → 1−. asymptotically (e.g. ρ → ∞) to one as opposed to the Since there are two other fixed points in the Gubser flow Bjorken model where a truly one-dimensional ODE was geometry, this condition guarantees that the flow trajecto- achieved via a similar trick in Ref. [25]. Therefore it is ries will not start in the other basins ⊂ XnBe; otherwise expected that the attractors for the Gubser flow in all the τ>−1 ϕ would always lie in XnBe. Hence any flow line hydrodynamical schemes are 1D nonplanar manifolds starting in the basin of attraction Be independently of their and correspondingly the basins of attraction are three whereabouts will always converge asymptotically to the dimensional. We analyze this below briefly in the attractor A at late times. context of dynamical systems for the IS theory but bear In general an attracting fixed point defining a basin of attraction for any (nonchaotic) model is referred to as the 10Two remarks are due here. First, we point out that in the dynamical equilibrium state of the system and for the Gubser flow geometry, the sink appears to be also located on the ˆ B Gubser system this is denoted by ðTe; π¯eÞ. From a differ- boundary of e which is related to the way the flow time is entiable geometry point of view, in the phase space of a 2D introduced in the state equations. We refer the reader to a different coordinate system used in Eq. (53) in which this sink is located autonomous dynamical system, one would expect the inside the basin. Second, the saddle point is a source of attractor A to be a manifold of codimension 1. For a 2D propulsion and this fact plays a role in determining the rate at nonautonomous dynamical system such as the one at hand, which the convergence of flow lines occurs.

044041-10 FAR-FROM-EQUILIBRIUM ATTRACTORS AND NONLINEAR … PHYS. REV. D 97, 044041 (2018)   in mind that a similar line of thought can be applied to 8 2 1 A∶ −2; − pffiffiffi ; − þ pffiffiffi ; any other theory as well. 3 5 3 3 5   8 2 1 B∶ −2; pffiffiffi ; − − pffiffiffi ; B. From the perspective of a 3D autonomous 3 5 3 3 5 dynamical system  pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 7 821 7 821 ∶ −2 − − In this section we study the linearization problem by C ; 5 15 ; 5 þ 15 : enhancing the 2D system to a 3D autonomous one, determine the stability conditions, and show that the One can then immediately see that A is a sink (all attractor is a nonplanar manifold of codimension 2. We eigenvalues are negative), and B, C have two positive will sketch the proof of the exponential asymptotic stability eigenvalues, thus making them saddle points, as expected of the steady state in Gubser flow and create a local and since the eigenvalues all are nonzero, the fixed points Lyapunov function to estimate the basin of attraction that is are hyperbolic. Hence, we can apply the Hartman-Grobman conjectured to be crucial in the study of an effective field (HG) theorem [73,74],11 that allows the local flow structure theory of stable hydrodynamical and nonhydrodynamical (phase-space portrait) near a hyperbolic fixed point to be modes. topologically equivalent to the flow diagram of its linear- ized system. For the Gubser flow in the IS theory, the flow 1. Linearization and exponential asymptotic stability diagram shown in Fig. 2 confirms the HG theorem in the – vicinity of all the fixed points. This figure also portrays the The nonautonomous system (34a) (34b) can easily be τ 0 7 extended to an autonomous system by considering the time state of flow lines on the time slice ¼ . of the three- τ as an additional variable. Therefore the IS theory of dimensional phase space. Using the HG theorem and the A Gubser flow is a truly 3D autonomous system of ODEs fact that is an exponentially asymptotically stable fixed given by point of Gubser flow, it is then easy to write down the Lyapunov exponents of phase-space variables along all the   trajectories converging to the attractor by the eigenvalues dTˆ 1 dπ¯ 4 1 1 ¼ Tˆðπ¯ − 2Þτ; ¼ − π¯2 τ − π¯ T;ˆ given above, which are dρ 3 dρ 3 5 c τ 2 1 8 d 2 λ − ffiffiffi λ − ffiffiffi λ −2 ¼ 1 − τ : ð47Þ Tˆ ¼ þ p ; π¯ ¼ p ; τ ¼ ; ð50Þ dρ 3 3 5 3 5

This is a polynomial vector field whose fixed points which matches the result previously obtained from the 2D ˆ analysis. The attractor may therefore be parametrized ðTc; π¯c; τcÞ are given by vectorially in the phase space spanned by ðu1; u2; u3Þ as pffiffiffi pffiffiffi R ρ −38c ; ; ; ; − ; ; ; ; : ρ0A ρ0 τ ρ0 ð /15 2 1Þ ð0 1/ 5 1Þ ð0 1/ 5 1Þ ρ d ðwð ÞÞ ð Þ e 0 u1 þð3AðwðρÞÞ þ 2Þu2 þ τρu3; ð51Þ ð48Þ for some constant ρ0. Evidently only for large ρ, τðρÞ tends Because of the symmetry ðT;ˆ π¯; τÞ → ð−T;ˆ π¯; −τÞ of to one exponentially faster than the other two converge to λ λ λ u the 3D problem (47)ffiffiffi , we are left withffiffiffi only three fixed the fixed point i.e., τ < π¯ < Tˆ such that 3 is just a point p p ρ → ∞ points A ¼ð0; 1/ 5; 1Þ, B ¼ð0; −1/ 5; −1Þ, and C ¼ in the phase space and Eq. (51) at lies on the plane u u ð−38c/15; 2; 1Þ. We now solve the linearized system spanned by f 1; 2g, and is well approximated by   around any fixed point, namely 1 ˆ λ ˆ ρ λ ρ T0e T u1 þ pffiffiffi − π¯0e π¯ u2 þ u3; ð52Þ 0 1 0 ˆ 1 ˆ 1ðπ¯−2Þ Tˆτ Tðπ¯−2Þ 5 ∂ρT τ B C B 3 3 3 C ˆ 2 @ ∂ρπ¯ A ¼ @ − π¯ − T − 8¯πτ 4 − 4¯π A given the initial conditions recovered from the asymptotics. c c 3 15 3 ∂ρτ 00−2τ ðTˆ ;π¯ ;τ Þ 0 1 c c c 11This mathematical theorem states that the behavior of a Tˆ − Tˆ nonlinear system of differential equations in a domain near its B c C @ π¯ − π¯ A hyperbolic equilibrium points is qualitatively the same as the × c ; ð49Þ behavior of the linearized version of these differential equations τ − τ near this fixed point. Hyperbolicity means that no eigenvalue of c the Jacobian matrix associated with the linearized dynamical system has a real part equal to zero. Therefore, one can use the and find the eigenvalues of the Jacobian matrix at every linearization of the original dynamical system to analyze its fixed point to be behavior around equilibria.

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Since the nonautonomous system (34a)–(34b) is expo- from the analysis above to be exponentially fast and nentially asymptotically autonomous for the attracting therefore one can expand around this fixed point by ˆ 1 fixed point, with the limiting functions giðT;π¯Þ being considering a variable of the type /t or any arbitrary the rhs of the two equations that resulted from positive real power of it. tanh2 ρ ∼ 1, Theorem 2.2 of Ref. [75] applies where solving for the attractor of either 2D or 3D systems would yield the 2. Local and global estimates of the basin same result. The main assumption is to suppose a 1-to-2 of attraction −2ρ transformation of the form t ¼e where for each value A path-integral analogy: Before going into any details, of ρ there is a pair of values for t, such that the steady state B we seek to motivate the reader to think about the question is not on the boundary of the basin of attraction, e. Then of why the basin of attraction and Lyapunov functions are ρ − 1 ˆ π¯ ≔−2 ¼ 2 log jtj in both cases. Also, let F3ðT; ;tÞ t extremely important concepts from a physical standpoint. where the index 3 denotes the rhs of the third equation in In the Feynman path-integral formalism of quantum field the 3D problem. We define for i ¼ 1, 2 (the rhs of the ith theory and quantum mechanics, one often encounters state equation) integrals of the sort 8   Z < ˆ π¯ 1ffiffi − 1 ≠ 0 fi T; þ p ; 2 log jtj ;t ϕ −S½ϕ ˆ 5 Z ¼ D e ð55Þ FiðT;π¯;tÞ¼   ð53Þ M : 1 ˆ π¯ pffiffi 0 gi T; þ 5 ;t¼ where S½ϕ is the action functional of the underlying theory that involves some field ϕ and the manifold M defines the where we have shifted π¯ for later purposes. With this new space of fields or paths over which the integral is time parametrization, Eq. (47) may be cast into the form performed. To keep the generality of the problem, we   complexify ϕ and thus S½ϕ is complex valued and M is a dTˆ 1 1 1 − jtj ¼ Tˆ π¯ − 2 þ pffiffiffi ; middle-dimensional manifold in the space of complex dρ 3 5 1 þjtj     fields. We will come to the dynamical system interpretation dπ¯ 4¯π 2 1 − jtj Tˆ 1 of this integral in a moment. But first, we have to note that ¼ − pffiffiffi þ π¯ − π¯ þ pffiffiffi ; dρ 3 5 1 þjtj c 5 the main problem of understanding what M is made of and finding it in a general field theory is extremely difficult. dt −2 Qualitatively, M is considered to be built out of the union of ρ ¼ t: ð54Þ d all the paths that are connected to a critical point (a stable fixed point of S½ϕ that satisfies some equation of motion We note that the Jacobian matrix of this 3D system, JacðFiÞ subject to the stationary action principle) which are the ˆ π¯ 0 has a block form at ðT; ; Þ and the origin is a fixed point, i.e. physically relevant paths (steepest descent paths) in the 0 0 0 0 Fið ; ; Þ¼ , and JacðgiÞð0;0Þ has negative eigenvalues. phase space or configuration space over which the field ϕ Since the original system is exponentially asymptotically takes values. So the most favored configuration is the one 0 0 0 autonomous, and gið ; Þ¼ the exponential asymptotic along which ReðS½ϕÞ increases as one approaches the stability immediately follows. critical point attached to it by some dynamical flow Last, a few remarks are due. There is a sense in which involving the action functional and an appropriate flow one must consider the aforementioned coordinate trans- time tf defined over the real line, for instance. This is formation inspired by the asymptotics of the series sol- described by an ODE of the form12 utions to the flow equations (34a)–(34b) mainly the study of resurgence properties and trans-series [26,27]. Focusing dϕ δS½ϕ ¼ ; ð56Þ on just a simple series solution is problematic on its own dt δϕ because of divergence issues but above all else lies the fact f that there is a challenge to pick a good expansion variable where the rhs is nothing but the variation of the action for the Gubser flow due to the peculiarities of de Sitter time. functional with respect to the dynamical field ϕ which now The natural choice for such a series expansion should at depends on tf and the bar represents complex conjugation. first glance be either tanh ρ or coth ρ, but they come with a “ ” ρ → ∞ The set of fixed points of this flow equation now caveat; they never grow bigger than 1 as and thus describes the saddle solutions to some equation of motion not are useful from the perspective of series asymptotics. It turns out that the exponential asymptotic stability of the 12 ϕ fixed point offers a suitable and rather natural choice for We note that even if is complexified, the number of degrees of freedom of the complex theory is the same as the original the expansion variable which will be briefly touched upon one on the manifold M, namely it is a middle-dimensional in Sec. IV B 2. It suffices to state that the rate at which the manifold in the complex space as dictated by the form of the flow converging flow trajectories approach the sink is known equation (56).

044041-12 FAR-FROM-EQUILIBRIUM ATTRACTORS AND NONLINEAR … PHYS. REV. D 97, 044041 (2018) governed by the action principle δS½ϕ/δϕ ¼ 0. This similarly to the negative of the real part of the action simple-looking flow equation resembles the more compli- functional decreasing on M due to stability conditions set cated dynamical system given by Eqs. (34a)–(34b) but the by the properties of Eq. (56).14 Given a global Lyapunov idea is that all the flow lines along which ReðS½ϕÞ → ∞ function, we therefore can write an effective action contribute to the construction of M13 where the stability functional for the Gubser flow in any theory15 → ∞ condition is guaranteed in Eq. (56) as tf since then a Z    fixed point is reached where ϕ remains constant for all dx 2 S ðx;cÞ ≔ dρ − Vðx;cÞ ; ð59Þ times thereafter. So M is a stable manifold of integration eff dρ shaped by the solutions to Eq. (56) that by construction is the manifold of steepest descent paths. We note that the where xðρÞ¼ðT;ˆ π¯;tÞ,andc is a “coupling constant.” initial conditions to solve this equation are picked by the Note that this effectively describes the underlying hydro- → −∞ convergence properties of Eq. (55) at tf . Namely, dynamical system as a theory of phase-space variables ϕ ¼ ϕ0 if ReðS½ϕ0Þ → ∞ at this limit; otherwise the path ˆ π¯; T;t taking values in Be—the basin of attraction for the integral would be divergent. The collective space of such stable fixed point. Finally the partition function can be “ ” initial values is called a good space denoted by Gc. formulated as Formally speaking, G ¼ ∪ G where in each good sub- c i i Z R space Gi the real part of the action functional remains x 2 − dρ d −V x;c ˆ π¯ ððdρÞ ð ÞÞ always positive. Hence the flow must begin from the points ZeffðcÞ¼ DTD Dte ; ð60Þ M in Gc where the paths can converge to the critical points along any direction in M. In a hydrodynamical system, we where M was defined in Eq. (57). This can be generalized can then regard M as a “Lefschetz thimble” made out of all to any theory of hydrodynamics which probes far-equi- the flow lines attached to the attracting fixed point for the librium aspects as well. In what follows, we create a local field ϕ in the path integral (55) starting at time ρ → −∞ and Lyapunov function and review a few techniques for the collective space Gc as the boundary of the basin of attraction. In other words, obtaining a global one that practically speaking is the ultimate goal of taking this approach. In an upcoming n o work, we will explore this effective partition function and ≔ ⋃ϕ ∶ϕ ϕ ρ → −∞ ∈ ∂B Gc i;0 i;0 ¼ ið Þ e ; to what extent it captures the properties of the original n i (relativistic) hydrodynamics. M ≔ ⋃ϕiðρÞ∶ϕiðρ → −∞Þ ∈ Gc; ϕiðρ → ∞Þ Local Lyapunov function: We showed in the previous i o subsection that the dynamical ODEs in Eq. (54) limit to a 2D autonomous system close to t 0, ¼ fixed point ∈ Be : ð57Þ ¼   ˆ dT 1 ˆ 1 There is in principle no universally well-established ¼ T π¯ − 2 þ pffiffiffi ; dρ 3 5 picture where we have an effective field theory approach     ˆ to hydrodynamics; rather we choose to go to the phase dπ¯ 4¯π 2 T 1 ¼ − pffiffiffi þ π¯ − π¯ þ pffiffiffi ; ð61Þ space of state variables and flow time to make our dρ 3 5 c 5 analogies with the picture given above. But what about the action functional in the phase space? It is kind of where (0,0) is an exponentially asymptotically stable obvious that the Lyapunov function(al) V that depends on fixed point similarly to (0,0,0) being the same for the the phase-space parameters of the underlying hydrody- 3D autonomous system. namical system acts like an effective action for all the V ˆ π¯ A local Lyapunov function locðT; ;tÞ can estimate the stable (and thus relevant) hydrodynamical modes. Once basin of attraction near the fixed point (0,0,0) of Eq. (54) V defined, is always positive definite, and more impor- that contains the origin inside its basin. Like the global tantly its derivative with respect to the flow time has to be V ˆ π¯ Lyapunov function, locðT; ;tÞ is a continuous positive- always nonpositive i.e. V ˆ π¯ definite scalar function locðT; ;tÞ defined on the set D ¼fjtj > 0; Tˆ ∈ R; π¯ ∈ Rg. V has continuous first- V ρ ≤ loc d /d 0; ð58Þ order partial derivatives at every point of D. Here we first

13Morse theory and its complex generalization, also called 14One can think of the effective action as a Morse-Bott Picard-Lefschetz theory are recent attempts toward understating functional, which satisfies dReð−SÞ/dρ ≤ 0. the means of building this M at least in calculable cases where it 15For a similar discussion on the relation between the effective is indeed of finite dimension such as quantum mechanics and potentials and Lyapunov potential functions based on gradient certain quantum field theories with nice properties [76–79]. flow equations, check Ref. [80].

044041-13 BEHTASH, CRUZ-CAMACHO, and MARTINEZ PHYS. REV. D 97, 044041 (2018) focus on a local construction of the basin of attraction, for some arbitrary W>0. Choosing W ¼ diagð1; 1; 1Þ,we V xT x where a local Lyapunov function loc ¼ P should exist solve this equation for P and insert the resulting matrix into T ˆ V ˆ π¯ such that x ¼ðT;π¯;tÞ and P satisfies the Lyapunov the formula for locðT; ;tÞ to obtain equation [81] V pffiffiffi pffiffiffi loc 2 2 2 ˆ 2 T − ¼ 29c ð57 5π¯ þ 76t þ 24ð10 þ 5ÞT Þ JacðFiÞð0;0;0ÞP þ PJacðFiÞð0;0;0Þ ¼ W; α 0 pffiffiffi 1 pffiffiffi pffiffiffi 1 342 10 − 7 5 π¯ ˆ 27 60 − 13 5 ˆ 2 3 ð1/ 5 − 2Þ 00 þ ð Þc T þ ð ÞT ; ð63Þ B ffiffiffi ffiffiffi C @ p p A JacðFiÞð0;0;0Þ ¼ −1/ð 5cÞ −8/ð3 5Þ 0 ; α 1 00−2 where ¼ 8816c2. As discussed before, along every flow V line in the basin of attraction, loc should effectively ð62Þ decrease with time; therefore

ffiffiffi ffiffiffi ffiffiffi ∂ρV p p p loc ¼ −58c2ð38t2ðjtjþ1Þþðjtj − 1Þ½2ðð10 þ 5Þπ¯ − 19ÞTˆ 2 − 19π ¯2ð 5π¯ þ 2ÞÞ − 57cπ¯ Tˆ½ð5ð5 5 − 3Þπ¯ þ 29Þjtj βðtÞ pffiffiffi pffiffiffi pffiffiffi pffiffiffi þð15 þ 4 5Þπ¯þ9Tˆ 2½ðð73 5 − 125Þπ¯ − 38 5 þ 133Þjtjþð60 5 − 65Þπ¯ < 0; ð64Þ

β 1 V where ðtÞ¼2204 2 1 is a positive function. This pro- f to be a Lyapunov function , one more condition comes c ðjtjþ Þ − V ρ vides the main restriction on the shape of the basin. from the fact that d /d has to be a SOS as well. V Another method that is similar in spirit to SOS pro- We note that the function loc defines a symmetric basin of attraction under t → −t which is consistent with gramming calls for a solution of the partial differential Proposition 2.9 of Ref. [75]. The two-time coordinate equation [75,83] patch ðT;ˆ π¯; e−ρÞ does in fact tell us that from the eyes of an observer sitting at the origin, there is a mirror symmetric copy of the attractor, say AR, with respect to the t ¼ 0 plane to which all the flow lines starting at t ≫ 0 will be seen to converge as well. This is simply a symmetry of the new coordinate system. The left copy AL is depicted in Fig. 3 along with the basin of attraction corresponding to the Lyapunov function in Eq. (63). Finding a Lyapunov function that captures the global basin of attraction for nonautonomous systems of higher dimensions (>1)isin general a hard problem and in most cases an analytic result cannot be obtained unless the dynamical system entails some nice properties due to hidden symmetries that could give rise to further simplifications. FIG. 3. The local basin of attraction for the 3D system (54) for Sum-of-squares (SOS) polynomials are another recent the Gubser flow in the IS theory using the Lyapunov function method developed based on global optimization for many (63). The boundary of the basin is topologically isomorphic to a applications including Lyapunov stability analysis and deformed 2-sphere which is locally a ρ ¼ const hypersurface. control theory [82]. This method attempts to find a sum For better visibility, the basin shape and size are set by the −100 V ρ ≤ V 5 of the form conditions

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FIG. 4. Basin of attraction for different initial times τ. We used as initial conditions a rectangular boundary in variables ðT;ˆ π¯Þ¼ ½1 × 10−4; 1 × ½−1.0; 0.4 and then chose an initial time, and let the 3D system evolve as Eq. (47). (a) In this plot the flow begins on the boundary of the basin of attraction at τ ¼ −1, and the surfaces made out of all the flow lines merge onto the attractor before finishing up at the stable fixed point. In plots (b), (c) and (d), there are two different behaviors observed for the initial conditions: one for when the flows start inside the basin of attraction, in which case the system evolves toward the stable fixed point asymptotically, and the other when the initial condition triggers the flows outside the basin of attraction, in which case the flow lines diverge. The separating blue flow lines determine a portion of the boundary of the basin of attraction.

V0ðt; xÞ¼−t2 − kxk2; ð66Þ (1) For any value of flow time, as long as the initial values initiate the flows in the basin of attraction, the where k::k shows the usual distance from the origin flow lines will always go toward the attractor. assuming that it defines a stable fixed point for the (2) The attractor is an invariant set of numbers toward dynamical system. The differentiation is implemented which the flow lines evolve inside the basin of with respect to a flow time ρ. It has been comprehensively attraction at late times. If there are no repelling fixed shown in the literature that in various (non)linear dynami- points around, with an underlying regular geometry, cal systems, global Lyapunov functions can be well the rate of convergence to the attractor for all the approximated by solving Eq. (66) using radial basis flows should be uniform. functions; see for instance Ref. [81]. We leave this to a (3) The flow line that begins at the saddle point located on future work and rather show a numerical plot of 2D the boundary of the basin, goes to the attractor at the surfaces in the basin of attraction for different initial fastest possible rate among all other flow lines because values of τ in Fig. 4. it gets propelled by the repulsive force of this point. ˆ Let us summarize the key take-home lessons learned in For a flow starting from ðTðρ0Þ; π¯ðρ0Þ; τðρ0ÞÞ, this this section: rate can be quantified by

044041-15 BEHTASH, CRUZ-CAMACHO, and MARTINEZ PHYS. REV. D 97, 044041 (2018) Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ A ˆ 2 2 2 2 d ðwÞ r ≔ ð∂ρTÞ þð∂ρπ¯Þ þð∂ρτÞ dρ; ð67Þ 3wðcoth ρ − 1 − AðwÞÞ þ HðAðwÞ;wÞ¼0; ρ0 dw ð68Þ which is literally the length of the curve of the velocity vector field or flow line. where the functional form of HðAðwÞ;wÞ depends on the (4) Out of all the flows starting at the same time slice, hydrodynamical model under consideration. For the hydro- say τ −1, the ones closer to the surrounding ¼ dynamical schemes studied in this work HðAðwÞ;wÞ takes saddle points will have more repelling force and three functional forms given by respectively thus a faster convergence. (5) It is not mentioned as often that the role of the basin 4 3AðwÞþ2 4 of attraction is important in determining the (multi) 3A 2 2 − HIS ¼ 3 ð ðwÞþ Þ þ 15 ; ð69aÞ stability and strength of the attractors as well as the cw   usefulness of the dynamical systems in considera- 4 1 10 4 tion even in the regimes far from (thermal) equilib- 3A 2 2 3A 2 − − HDNMR ¼ 3 ð ðwÞþ Þ þð ðwÞþ Þ 7 15 ; rium. So knowing the topology and size of the basin cw of attraction is necessary for the study of hydrody- ð69bÞ namical flows and the search for attractors per se in a   dynamical system with a stable fixed point and some 4 1 4 H ¼ ð3AðwÞþ2Þ2 þð3AðwÞþ2Þ − unstable fixed points would not be illuminating. aHydro 3 cw 3 – Things evolve toward stability either quickly or 5 3 slowly—as long as they are initiated with values − þ Fð3AðwÞþ2Þ: ð69cÞ in the basin of attraction. 12 4 (6) The attractor line has the fastest rate of convergence due to the propulsion of the saddle point it starts These expressions were determined using the conserva- from and therefore naturally there is always a fast tion law (12) together with Eqs. (13a) and (13b) for IS and convergence of modes nearby even in the far-from- DNMR respectively while for anisotropic hydrodynamics equilibrium regime of hydrodynamics. we considered Eqs. (23) and (25). We note that for aHydro (7) Any effective theory of hydrodynamics can be it is necessary to rewrite the function (29) in terms of written as a simple kinetic term with a Lyapunov AðwÞ. potential functional that describes the relevant physi- When applying the slow-roll-down approximation cal modes that only belong to the basin of attraction. dA/dw ¼ 0inEq.(68) one needs to find the roots In the effective field theory language, everything of HðAðwÞ;wÞ ≡ 0. We take the late-time or asymptotic outside the basin of attraction is integrated out. limit tanh2 ρ ∼ 1 which was explained before and it poses no problem in the universality of the attractor due IV. UNIVERSAL ASYMPTOTIC ATTRACTORS to the exponentially fast convergence of flow lines FOR DIFFERENT DYNAMICAL MODELS toward it at late times. In the case of IS and DNMR this The relaxation equations of the different components of constraint gives two different solutions so one chooses A the energy-momentum tensor derived from a particular only the stable one þðwÞ [25,26], which are respec- microscopic theory do not lead necessarily to the same tively given by attractor. In this section we present numerical calculations "  of the attractors of different hydrodynamical models and 1 3 A ¼ − 16 þ the exact solution of the RTA Boltzmann equation. þ 24 cw In order to determine the attractors we follow closely the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  # procedure outlined in the work of Heller and Spalinski [25] 3 2 76 2 þ 16 þ − 48 þ ; ð70aÞ which was based on the slow-roll-down approximation cw 15 cw [84].16 The reader should bear in mind that the attractors calculated in this section are asymptotical since the basin of "  attraction is three dimensional as we argued in the previous 1 82 3 A ¼ − þ section. þ 24 7 cw The equations of IS, DNMR and aHydro can be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  # combined into a unique ODE for the function AðwÞ defined 82 3 2 232 2 þ þ − 48 þ : ð70bÞ in Eq. (45), which reads as 7 cw 105 cw

16This is more rigorously equivalent to walking on a null line The initial condition for solving the differential in the vicinity of a stable fixed point. Eq. (68) for IS and DNMR is obtained by evaluating

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FIG. 5. Attractors for different theories (solid red line) and numerical solutions for a large set of initial conditions (solid gray lines). The (steady state) nonequilibrium attractors shown in this figure correspond to IS (top left panel), DNMR (top right panel), aHydro (bottom left panel) and the exact solution of the Gubser flow (bottom right panel). In each case we use c ¼ 15/ð4πÞ.

A ≡ A 17 limw→−∞ þðwÞ i. Afterwards, as mentioned by following the technique explained in Ref. [35].For before, one simply solves Eq. (68) while taking completeness we briefly explain it in Appendix B. When ρ → −∞ (and thus coth2 ρ − 1 → 0). For the case of solving it numerically we used ρ0 ¼ −10, which is good aHydro the initial condition is determined by finding enough for our purposes because this value avoids unphys- numerically the roots of HðAðwÞ;wÞ [Eq. (69c)]which ical behavior e.g. negative temperatures [13,85]. In Fig. 5 the asymptotic attractors (late-time tail of the gives Ai ≈−0.75. In the next subsection we discuss the numerical solutions of Eq. (68) for each approximation solid red line) of the IS (top left panel), DNMR (top right scheme. panel), anisotropic hydrodynamics (bottom left panel) and the exact solution of the Gubser flow (bottom right panel) for a variety of initial conditions (gray lines) are shown. In A. Numerical results each plot we chose c ¼ 5η/s with η/s ¼ 3/ð4πÞ. The set of We are now ready to discuss the numerical results for the initial conditions were chosen by allowing the initial attractors (late-time asymptotic behavior) for each trunca- condition of the effective shear to be either prolate ˆ ˆ tion scheme together with the exact attractor of the Gubser (π¯ < 0 and thus PL < P⊥), or oblate (π¯ > 0 and thus ˆ ˆ solution (31). The exact location of the attractor was found PL > P⊥). The former configuration corresponds to Ai < −2/3 while the latter indicates Ai > −2/3. The initial A 17Alternatively, one can also predict the value of the initial values i of the different asymptotic attractors (late-time A → −∞ tail of the solid red line) in Fig. 5 are always below A < condition for þðw Þ from the method discussed in i Sec. III. In this case one equates the value of π¯ðρ → −∞Þ at −2/3 so that the steady-state attractor of fluids undergoing the stable fixed point to the conservation law (12). For instance, Gubser flow corresponds to a prolate configuration. This 15 π π¯ ρ → −∞ for c ¼ /ð4 Þ the values ofpffiffiffið Þ at thep stableffiffi fixed result is valid independently of the hydrodynamical model points for IS and DNMR are 1/ 5 and 15/28 − 1/28 ð1909/5Þ, and is in agreement with the methods discussed in Sec. III A ρ → −∞ respectively, so þð Þ in turn takes the approximate −0 8157 −0 7207 since the saddle fixed points are located precisely when values . and . for the former and the latter π¯ 0 hydrodynamics model. These numbers coincide exactly with jρ→−∞ < . We also notice that independently of the theory, the w → −∞ limit of Eqs. (70a)–(70b). prolate configurations merge faster to their corresponding

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-0.25 accuracy. Now, the best theory to describe the exact -0.3 attractor shown Fig. 6 is aHydro. A closer look shows -0.4 -0.35 that all the hydrodynamical models are not able to match -0.5 the exact result in the small w region (−0.8 ≲ w ≲ 2)asis

-0.45 -0.6 shown in the inset of Fig. 6. However, the numerical

-0.7 difference between IS and aHydro with respect to the exact result is no larger than 4% in this w interval while DNMR -0.55 -0.8 -3 -1 1 3 5 7 9 deviates entirely in this regime of w. In the large or intermediate regime, on the other hand, we verify numeri- -0.65 cally that the largest numerical deviation between the aHydro attractor and the exact one does not exceed IS -0.75 DNMR 0.06%. The numerical results presented here provide aHydro Exact RTA conclusive proof that aHydro resums effectively the -0.85 Knudsen and inverse Reynolds numbers to all orders -140 -120 -100 -80 -60 -40 -20 0 20 40 60 independent of the initial conditions. We point out again that the notion of “attractor solution” is ill defined and one FIG. 6. Asymptotic attractors of the IS (dotted green lines), should not care about what occurs in the mid-range or DNMR (dash-dotted blue lines), aHydro (short-dashed orange early-time regimes of w because the attractor is actually a lines) and exact Gubser solution (solid black line). For all cases statement about the late-time asymptotics of the flow lines. 15 π we use c ¼ /ð4 Þ. We can only say that the attractor solution is just a solution to some system of ODEs with a given initial value that is attractors than the oblate ones. Furthermore, the positive- located exactly at the saddle point on the boundary of the definite condition of the transverse and longitudinal pres- basin of attraction. But because there is one unstable and sures implies that −3/4 < AðwÞ < −1/2. The positivity of two stable directions at this point (one of the eigenvalues of the pressures is satisfied only by aHydro and the exact the Jacobian matrix being positive implies an unstable — — Gubser solution and thus one cannot initialize AðwÞ below direction), then as it is metastable the saddle point will these values. This statement was tested numerically and initiate a flow inside the basin of attraction and finally the explains why there are no gray lines below the early-time attractor will absorb it. We point out that this flow line is line of the attractor in the bottom panels of Fig. 5. However, the fastest to converge to the stable fixed point and thus the for the IS and DNMR evolution equations this condition can attractor. In general this is an invariant set of numbers that be slightly broken by initializing AðwÞ below the attractor flow lines converge to at very large w and as can be seen in as is shown in the top panels of Fig. 5. This indicates that the Fig. 6, even for w>0 there is no separation between flow basin of attraction of IS and DNMR theories is larger than lines due to the exponentially fast convergence. This is aHydro and the exact solution while being physically much better than what we could have asked for from the 2 invalid. If one restricts IS and DNMR to satisfy the approximation tanh ρ ∼ 1 that was applied to get AðwÞ positivity condition of the pressures, it would physically in Eq. (68). shrink the phase space of initial conditions and thus, the basin of attraction [86]. B. Asymptotic perturbative series expansion A natural question which arises in our context is, what is The numerical comparisons between different models the best model that matches the underlying exact micro- discussed previously demonstrated that the IS and DNMR scopic kinetic theory? We answer this question by plotting theories cannot describe the exact steady-state attractor. in Fig. 6 the attractors of different hydrodynamical theories This failure is somehow expected since both theories are together with the one obtained for the exact Gubser solution derived from an asymptotic series expansion of the dis- (31). First, we observe that none of the truncated approxi- tribution function. Here we show strong evidence of the mated schemes—DNMR and IS—are able to be in good divergence of this series by numerically solving the agreement with the exact attractor over the entire w regime recursion relations among the expansion coefficients of studied here. One might be at first surprised that none of the an asymptotic series ansatz. These recursion relations are hydrodynamical truncation schemes do work even at large derived analytically from the underlying hydrodynamical w when the system supposedly reaches its thermal state. equations of motion. We briefly comment on how to fix the However, the Gubser flow does not reach this state divergences by using resurgence techniques. asymptotically since the expansion rate θˆ ¼ 2 tanh ρ satu- rates at large ρ without vanishing exactly. Among these two hydrodynamical truncation schemes, we find the IS to 1. Divergence of IS and DNMR theories be closer to the numerical values of the exact attractor than Since w ∈ ð−∞; ∞Þ, one can propose an asymptotic DNMR albeit still unable to match it to high numerical series ansatz of the form

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that in the large-ρþ limit the DNMR solution does a poorer job than IS when comparing the predictions of both theories with the RTA exact solution (31).

2. Resurgence to the rescue The true nature of the attractor from the perspective of solving for a series solution would be unveiled by studying the resurgent asymptotic expansions around this solution which mimics the instanton corrections on top of perturbation theory in quantum field theories. This can be packaged into a formal exponential series solution known as a trans-series which accounts for all sorts of A 1/n FIG. 7. j nj as a function of n for both IS (black) and DNMR corrections such as exponential, logarithmic, etc. [88].For 15 π (red) theories. We use c ¼ /ð4 Þ. the system of Eqs. (34a)–(34a) with the substitution τ ¼ ð1 − e2ρÞ/ð1 þ e2ρÞ an analytic solution of this sort can be X∞ written as [89] −n AðwÞ¼ Anw ; ð71Þ 0 ρ n¼ ˆ ρ ˆ ð0;0Þ ρ 1 0 m1ρ −jλ ˆ je ˆ ð1;0Þ ρ Tðe Þ¼T ðe Þþσ1σ2e e T T ðe Þ ρ 0 1 m2ρ −jλπ¯ je ˆ ð0;1Þ ρ that is claimed to solve Eq. (68) in the asymptotic limit þ σ1σ2e e T ðe Þþ ρ → ∞ with HðAðwÞ;wÞ given by Eqs. (69a) and (69b). X∞ ⃗ ρ n1 n2 m⃗·n⃗ρ −n⃗·λe ˆ ðn1;n2Þ ρ After equating this ansatz to Eq. (68) one obtains the ¼ σ1 σ2 e e T ðe Þ; ð74Þ following recursive relations for the kth coefficient (k ≥ 2) n1;n2¼0 of the IS and DNMR series, respectively: where eρ is a good variable for the trans-series since at large ˆ −λ ˆ ρ −λ ˆ Xk A 16A ρ, w ∼ 1/TðρÞ ∼ coshðρÞ T ∼ ðe Þ T from the asymptotic 4 A A k−1 k 0 pffiffiffi ðn þ Þ n k−n þ þ ¼ ; ð72aÞ ⃗ 1 1ffiffi c 3 solution (36), with λ ¼ðjλ ˆ j; jλπ¯jÞ ¼ ð3 ð2 − p Þ; 8/ð3 5ÞÞ n¼0 T 5 being the absolute value of the Lyapunov exponents of Xk trajectories converging to the attracting fixed point. Here, Ak−1 82Ak ðn þ 4ÞAnAk−n þ þ ¼ 0: ð72bÞ c 21 X∞ n¼0 ⃗ ˆ ðn1;n2Þ −n⃗·ðm⃗þβÞ ˆ ðn1;n2Þ −lρ T ¼ e Tl e ; ð75Þ pffiffiffi pffiffi A 1 5 − 10 A − 9þ2 5 l¼0 For the IS 0 ¼ 15 ð Þ and ffiffiffiffiffiffiffiffiffiffi1 ¼ 61cffiffiffi while 1 p p A0 − pffiffi 1909 41 5 ⃗ for the DNMR ¼ 84 5 ð þ Þ and is the formal power series and m⃗ m1;m2 1;1 −Int β pffiffiffiffiffiffiffi ¼ð Þ¼ð Þ ½ A − 1266þ11 9545 “ ” β⃗ 1 ¼ 9139c . The truncation of this power series with Int½: meaning the integer part of where is a at n ¼ 30 is sufficient for the purpose of convergence tests constant real vector field whose components are the in general. In Fig. 7 we present the numerical solutions of coefficients of the term proportional to ðe−ρT;eˆ −ρπ¯Þ after – the recursive Eqs. (72a) (72b) for both IS and DNMR linearization. σ1;2 ∈ C are the trans-series expansion theories when c ¼ 15/ð4πÞ. From this figure we observe parameters and the real part of σ1 imposes the initial that for both of these hydrodynamical models condition for the temperature. Similarly, for π¯ we propose a trans-series of the form 1/n 1/n 1 < lim sup ðAnÞ < lim sup ðAnÞ ; ð73Þ →30 n→∞ ρ n ρ ð0;0Þ ρ 1 0 m1ρ −jλ ˆ je ð1;0Þ ρ π¯ðe Þ¼π¯ ðe Þþσ1σ2e e T π¯ ðe Þ ρ A 0 1 m2ρ −jλπ¯ je ð0;1Þ ρ so the root test [87] fails for the ansatz of ðwÞ [Eq. (71)] þ σ1σ2e e π¯ ðe Þþ and thus, the asymptotic series expansions of the IS and X∞ ⃗ ρ n1 n2 m⃗·n⃗ρ −n⃗·λe ðn1;n2Þ ρ DNMR theories are divergent. Figure 7 also indicates that ¼ σ1 σ2 e e π¯ ðe Þ: ð76Þ in this asymptotic regime the divergence in the DNMR case n1;n2¼0 is more severe. This is due to the new term in Eq. (69b) which survives in the large-w limit instead of converging to The real part of σ2 ∈ C encodes the initial data for π¯ in zero and thus any divergence in AðwÞ in the IS theory will solving for the full set of trans-series solutions. be magnified in the DNMR theory. This result also But what about the imaginary parts? In reality, the confirms earlier numerical studies of the Gubser flow leading large-order terms in both factorially divergent (see for instance Fig. 3 in Ref. [13]) where it was found series π¯ð0;0Þ and Tˆ ð0;0Þ,

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0 0 ðl − 1 þ m1Þ! 0 0 ðl − 1 þ m2Þ! V. CONCLUSIONS Tˆ ð ; Þ ∝ ; π¯ð ; Þ ∝ ð77Þ l λ lþm1 l λ lþm2 j Tˆ j j π¯j The properties of the attractors of different hydrody- namical systems undergoing Gubser flow were studied where l → ∞, can be transformed into functions with within relativistic kinetic theory. Our work extended singularities in the complex plane that need to be dodged previous studies of attractors by incorporating techniques via careful contour deformations. We introduce the Borel and tools of nonlinear dynamical systems. We extensively transform, discussed the application of these methods to the IS evolution equations to investigate the stability properties ∞ 0 0 ∞ 0 0 of the fixed points, the flow lines around those and the X Tˆ ð ; Þsl X π¯ð ; Þsl B½Tˆ ð0;0ÞðsÞ¼ l ; B½π¯ð0;0ÞðsÞ¼ l Lyapunov exponents. These techniques were proven to be l! l! l¼0 l¼0 strikingly useful in the understanding of the reason behind ð78Þ the impossibility of reducing the 2D system of ODEs of each hydrodynamical scheme into a single one universally as it was done in the Bjorken case [25]. It was shown that that has a finite radius of convergence by construction. To this nonreduction is intrinsically related to the dimension- capture the divergence of the original series as a tangible ality of the basin of attraction and that the tanh ρ also acts as object, one performs a Borel summation of the series in an independent variable of ρ that is responsible for the extra Eq. (78), of the preferred form dimension of the phase space compared to the Bjorken Z case. Inspired by the exponential asymptotic stability ∞þiϵ T ˆ ð0;0Þ ρ −eρsB ˆ ð0;0Þ and resurgent trans-series-type arguments, we defined a ϵ½T ðe Þ¼ e ½T ðsÞds; −2ρ Z0 natural time variable, t ¼e , which was expounded in ∞ ϵ þi ρ Sec. III B 2. In this coordinate system, the observer sees ð0;0Þ ρ −e s ð0;0Þ T ϵ½π¯ ðe Þ¼ e B½π¯ ðsÞds; ð79Þ → ∞ 0 two copies of the same flow starting at t that approach the point (0,0,0) which is now the stable fixed point of the system, hence making the basin symmetric which are nothing but the inverse Laplace transforms of the under t → −t. This method is a very well-known trick used Borel functions. Since there are in general singularities on in usually exponentially asymptotically stable systems to the real line in the Borel plane (s plane) that capture the connect the basin of attraction in nonautonomous systems divergence of the original series, there is a slight shift of the to that of autonomous ones as well as compactifying the integration contour ðϵ ¼þ0Þ or ðϵ ¼ −0Þ and depending on how this is qualitatively done, we would end up with a basin in case the time dependency is in the form of an discontinuity along the contour of integration that by the unbounded function. It also brings the fixed point to the Cauchy residue theorem would generate a pure imaginary origin which is crucial for the construction of Lyapunov part once we jump from the þi0 to −i0 contour or vice functions. This helped us build such a function for the IS versa. This is related to the fact that the real line is simply a theory and discuss the shape of the basin of attraction at “Stokes line,” and the jump associated with the disconti- least locally near the steady-state equilibrium. This notion nuity across it, is called the “Stokes jump” performed by a of the Lyapunov function from dynamical systems was shown to be related to an effective-action description of Stokes constant iS1 with S1 being real. These jumps are calculated via the formula hydrodynamics motivated by Picard-Lefschetz theory. The attractors of the IS, DNMR and aHydro models were 8 obtained via the slow-roll-down approximation while the > σ⃗− σ⃗ −0 ϵ 0 <> I ¼ Ið Þ for < ; attractor associated to the Gubser RTA solution was σ⃗ϵ σ⃗0 σ⃗ −0 1 ˆ ϵ 0 determined through Romatschke’s method [35]. From ð ÞI ¼ I ¼ Ið Þþ2 S1e1 for ¼ ; ð80Þ > the numerical comparisons between the attractors of differ- : σ⃗þ σ⃗ −0 ˆ ϵ 0 I ¼ Ið ÞþS1e1 for > ent theories with the exact solution we conclude that (a) hydrodynamical models based on an asymptotic series expansion of the distribution function (i.e., IS and DNMR) where eˆ1 ¼ð1; 0Þ and σ⃗I ¼ðImσ1; Imσ2Þ, which was obtained originally by a “balanced average” summation are unable to provide a quantitative description of the of the formal divergent series [89]. Therefore, the reality attractor of the exact solution and, (b) the best agreement condition on the trans-series exactly on the positive real with the exact attractor is achieved by aHydro up to high axis would imply that numerical accuracy. For the IS and DNMR approximation schemes we showed that their corresponding asymptotic   1 solutions to the respective differential equations are diver- σ⃗0 0 ⇒ σ⃗ 0 gent. The origin of the divergences and how to resum them I ¼ I ¼ S1; : ð81Þ 2 was briefly discussed by applying resurgence techniques.

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The success of aHydro to describe the asymptotic Energy Scan Theory) DOE Topical Collaboration. A. B.’s attractor of the exact Gubser solution demonstrates that portion of the research was supported in part by the this theory is able to take into account both, large inho- National Science Foundation under Grant No. NSF mogeneities in the fluid due to collisions (quantified by the PHY-1125915. A. B. thanks the local organizers of the Knudsen number) as well as big spacetime inhomogeneities “Resurgent Asymptotics in Physics and Mathematics” of the macroscopic fluid variables (quantified by the inverse workshop at KITP for the warm hospitality. Reynolds number). As a matter of fact, the match between the aHydro attractor and the exact one demonstrates that APPENDIX A: ANISOTROPIC INTEGRALS −1 aHydro resums the Kn and Re to all orders. This ˆ resummation is carried out by reorganizing the asymptotic Here we calculate the anisotropic integrals Inlq that expansion series in a nonperturbative way by including the appear in this paper. First we define largest momentum-space anisotropies present in the plasma ˆ ˆ n−l−2q ˆ l ˆ μ ν q I ðΛ; ξÞ¼hð−uˆ · pˆ Þ ðl · pˆ Þ ðΞμνpˆ pˆ Þ i into the leading-order anisotropic distribution function. nlq Z   a pˆ 2 q The very common techniques in the context of nonlinear ˆ ρ n−l−2q ˆ l Ω ¼ ðp Þ pη 2 fa: ðA1Þ dynamical systems presented in this work and applied to pˆ cosh ρ hydrodynamics, can be extended to study a long list of properties of other relevant more complex physical systems By considering the change of variables than the one studied here. This list may include an pˆ θ investigation of hydrodynamization processes and the ¼ λ sin α cos β; ðA2aÞ dynamics of attractors for more general nonlinear colli- cosh ρ sional kernels [7,22,90], holographic models [8,14–18], pˆ ϕ spatially nonhomogeneous expanding fluids [39] and non- ¼ λ sin α sin β; ðA2bÞ relativistic systems [51,52]. On the other hand, it is also cosh ρ sin θ interesting to investigate the rich structure and topology of ˆ λ 1 ξ −1/2 α the basin of attractors in turbulent flows and other chaotic pη ¼ ð þ Þ cos ; ðA2cÞ systems of interest. On a more theoretical subject, the possibility of formulating effective actions for hydrody- one is able to factorize the integral (A1) as namics by exploring the analogy between the steepest ˆ Λˆ ξ Jˆ Λˆ Rˆ ξ descent directions in the path integrals and the flow lines Inlqð ; Þ¼ nð Þ nlqð Þ; ðA3Þ starting at the boundary of the basin of attraction for the attractors opens a new perspective to reformulate this old where Z problem. Moreover, the issues addressed in this work shed ∞ dλ ˆ n 1 ! Jˆ Λˆ λnþ1 −λ/Λ ð þ Þ Λˆ nþ2 more light on new questions that remain to be answered nð Þ¼ 2 e ¼ 2 ; ðA4Þ within the aHydro framework. For instance, in the resur- 0 2π 2π Z gence program, the attractor is understood as the leading- 1 1 order asymptotic trans-series [43,91]. This mathematical Rˆ ξ dx 1 − x2 qxl nlqð Þ¼ ðn−2qÞ/2 ð Þ statement together with our results suggest a highly non- 2ð1 þ ξÞ −1 trivial relation between the nonperturbative resummation of × ½ð1 þ ξÞð1 − x2Þþx2ðn−l−2q−1Þ/2: ðA5Þ large Knudsen and inverse Reynolds numbers carried out ˆ by aHydro and a certain class of solutions of the nonlinear The explicit forms of the functions Rnlq used in this paper Boltzmann equation that can be expressed as a trans-series. are given by We leave these matters to future works.  pffiffiffi ˆ 1 1 arctan ξ ACKNOWLEDGMENTS R200ðξÞ¼ þ pffiffiffi ; ðA6aÞ 2 1 þ ξ ξ We would like to thank Gökçe Başar, Gerald Dunne,  pffiffiffi ˆ 1 1 arctan ξ Behnam Kia, Thomas Schäfer and Mithat Ünsal for useful R220ðξÞ¼ − þ pffiffiffi ; ðA6bÞ discussions. We would also like to specially thank David 2ξ 1 þ ξ ξ Sauzin and Ovidiu Costin for very useful discussions and  pffiffiffi helpful comments. C. N. C. C. would like to thank E. C. ˆ 1 3 þ 2ξ arctan ξ R240ðξÞ¼ − 3 pffiffiffi : ðA6cÞ Pinilla for her continuous support and useful discussions. 2ξ2 1 þ ξ ξ We thank Michal Heller for reading the draft and his ˆeq comments/suggestions. A. B. and M. M. are supported in The moments Inlq associated with the equilibrium distri- part by the US Department of Energy Grant No. DE-FG02- bution function are obtained by considering the ξ → 0 limit 03ER41260 and M. M. is supported by the BEST (Beam of Eq. (A3):

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which is nothing but the equation for the A null line. As we Iˆeq ðTˆÞ ≡ limIˆ ðΛˆ ; ξÞ¼Iˆ ðT;ˆ 0Þ: ðA7Þ nlq ξ→0 nlq nlq pointed out in Secs. III and IV one eliminates the ρ dependence by considering the asymptotic behavior at ρ → −∞. For numerical purposes it is enough to take APPENDIX B: ATTRACTOR OF THE RTA ρ0 ¼ −10 and thus limρ→−10 tanh ρ ≈−1. By doing this the BOLTZMANN EQUATION FOR terms entering in the constraint (B1) are given by THE GUBSER FLOW ϵˆ ˆ Λˆ ξ The attractor for the exact solution (31) was found jρ¼−10;ξ¼ξ0 ¼ I200ð 0; 0Þ; by considering Romatschke’s technique [35]. For the ˆ ˆ ˆ ˆ ∂ρϵˆjρ −10 ξ ξ ¼ 3I200ðΛ0; ξ0Þ − I220ðΛ0; ξ0Þ; Gubser case the slow-roll condition, cf. Refs. [25,35,84], ¼ ; ¼ 0 ∂2ϵˆ 9 ˆ Λˆ ξ − 4ˆ Λˆ ξ dAðwÞ/dw ¼ 0 gives ρ jρ¼−10;ξ¼ξ0 ¼ I200ð 0; 0Þ I220ð 0; 0Þ Rˆ ξ 1/4Λˆ   ˆ Λˆ ξ ½ 200ð 0Þ 0 1 1 1 ∂ ϵˆ þ I240ð 0; 0Þþ 2 2 ρ c ðϵˆ∂ρϵˆ − ð∂ρϵˆÞ Þ − ¼ 0; 4 ϵˆ2 ρ 2ρ ϵˆ ˆ Λˆ ξ − ˆeq Rˆ ξ 1/4Λˆ tanh sinh ρ¼ρ0;ξ¼ξ0 × ½I220ð 0; 0Þ I220ð½ 200ð 0Þ 0Þ: ðB1Þ ðB2Þ

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