Adiabatic hydrodynamization in rapidly-expanding quark-gluon plasma

We propose a new scenario characterizing the transition of the quark-gluon plasma (QGP) produced in heavy-ion collisions from a highly non-equilibrium state at early times toward a fluid described by hydrodynamics at late times. In this scenario, the bulk evolution is governed by a set of slow modes, after an emergent time scale $\tau_{{\rm Redu}}$ when the number of modes that govern the bulk evolution of the system is reduced. These slow modes are"pre-hydrodynamic"in the sense that they are initially distinct from, but evolve continuously into, hydrodynamic modes in hydrodynamic limit. This picture is analogous to the evolution of a quantum mechanical system that is governed by the instantaneous ground states under adiabatic evolution, and will be referred to as"adiabatic hydrodynamization". We shall illustrate adiabatic hydrodynamization using a kinetic description of weakly-coupled Bjorken expanding plasma. We first show the emergence of $\tau_{{\rm Redu}}$ due to the longitudinal expansion. We explicitly identify the pre-hydrodynamic modes for a class of collision integrals and find that they represent the angular distribution (in momentum space) of those gluons that carry most of the energy. We use the relaxation time approximation for the collision integral to show quantitatively that the full kinetic theory evolution is indeed dominated by pre-hydrodynamic modes. We elaborate on the criterion for the dominance of pre-hydrodynamic modes and argue that the rapidly-expanding QGP could meet this criterion. Based on this discussion, we speculate that adiabatic hydrodynamization may describe the pre-equilibrium behavior of the QGP produced in heavy-ion collisions.

Hydrodynamics describes the real-time dynamics of a broad class of interacting many-body systems in the long time and long wavelength limit. In this limit, most degrees of freedom become irrelevant since they relax on short time scales. The surviving slow dynamical variables, or "hydrodynamic modes", are those associated with conserved densities such as the energy density. Hydrodynamic modelling has seen remarkable success at describing varied and non-trivial results of heavy-ion collision experiments (see Ref. [1] for a concise review). This in turn raises the important question of how the system approaches a state dominated by hydrodynamic modes, namely how "hydrodynamization" occurs in the aftermath of a heavy-ion collision (cf. [2][3][4] for a recent review).
In this letter, we theorize a new scenario for the process of hydrodynamization with the following defining attribute: during the interval τ Redu < τ < τ Hydro , the bulk evolution is governed by a set of slow modes that are "pre-hydrodynamic" in the sense that they are distinct from hydrodynamic modes but evolve gradually into them around the time τ Hydro . As a premise of this picture, we assume the emergence of a time scale τ Redu < τ Hydro around which the degrees of freedom required to describe the bulk properties of the system are reduced (see more below).
The pre-hydrodynamic modes in the preceding scenario are the modes with the slowest rate of change at each instant in the pre-hydrodynamic evolution, under the assumption that they remain gapped from faster modes. They are closely analogous to the instantaneous ground states of a time-dependent Hamiltonian in quantum mechanics. Since near thermal equilibrium the hydrodynamic modes are the slowest modes, the pre-hydrodynamic modes are a natural off-equilibrium generalization of the hydrodynamic modes. If a timedependent and gapped quantum-mechanical system is prepared in its ground state, it will remain in the instantaneous ground state under adiabatic evolution of the Hamiltonian. We will thus refer to situations where the approach to hydrodynamics is governed first by the evolution of pre-hydrodynamic modes as "adiabatic hydrodynamization" (AH).
We will illustrate AH in a kinetic description of weaklycoupled Bjorken-expanding plasma. We explicitly identify the pre-hydrodynamic modes as the instantaneous ground state modes of a non-Hermitian matrix describing the evolution of bulk quantities from the kinetic equation with a class of collision integrals. Physically, these modes represent the angular distribution in momentum space of the gluons that carry most of the energy of the system. We then demonstrate the emergence of τ Redu induced by the fast longitudinal expansion and show that τ Redu is parametrically smaller than τ C . This is due to the separation of scales between the initial time τ I when the kinetic description becomes applicable and the typical collision time τ C for QGP in the weak-coupling regime. Because of the hierarchy τ Redu τ C τ Hydro , the prehistory of hydrodynamics is (almost) the history of prehydrodynamic modes within AH.

arXiv:1910.00021v1 [nucl-th] 30 Sep 2019
An important implication of AH is that the macroscopic properties of the medium during the prehydrodynamic stage are insensitive to both the initial conditions and the details of the expansion history, and instead are determined predominantly by the features of the pre-hydrodynamic modes. In particular, the most important quantity characterizing the bulk evolution of a plasma undergoing Bjorken expansion is the percentage rate of change of the energy density where y ≡ log(τ /τ I ) plays the role of a time variable. We shall show that g(y) is related to the eigenvalue E 0 (y) of the pre-hydrodynamic mode if AH applies, namely We consider the extensively-studied relaxation time approximation (RTA) of the kinetic equation [5][6][7][8][9][10] and confirm quantitatively that Eq. (2) holds, demonstrating that hydrodynamization in this model is an example of AH.
Because of the expansion, the criterion for the dominance of pre-hydrodynamic modes is not that the excited states have decayed, but rather that transitions to the excited states are suppressed. In the absence of better terminology, throughout this manuscript we will use "adiabaticity" as a synonym for the suppression of these transitions. This is consistent with the modern use of this terminology in quantum mechanics (c.f. Ref. [11]). We will show that the regime where this generalized notion of adiabaticity may not apply is parametrically narrow according to the scenario of bottom-up thermalization for weakly-coupled QGP [12]. Although our analysis relies on the smallness of α s , we hope that many qualitative features of AH may nonetheless be present in the QGP created in heavy-ion collisions.
The key premise of this paradigm is that hydrodynamic modes dominate the bulk evolution, and consequently that hydrodynamics is applicable, even when the system is far from equilibrium [13]. The difference between this paradigm and AH is that the dominant slow modes for systems undergoing AH are pre-hydrodynamic mode(s), which can be qualitatively distinct from hydrodynamic modes. The slow modes of a system generally depend on the state of the medium under consideration, and therefore it is unsurprising that the slow modes in a far-fromequilibrium system are generically different from the hydrodynamic modes. A useful example is the low-energy collective excitations in a normal Fermi liquid. When the frequency of the distribution function variation, an analog of expansion rate, is much larger than the collision rate and thermal equilibrium is not established in each volume element, the slow modes are zero sound modes which have different physical characteristics than ordinary sound. We show that RTA kinetic theory is an example where pre-hydrodynamic and hydrodynamic modes are qualitatively different, and the bulk evolution of the system is dominated by the pre-hydrodynamic modes.
In the modern view, hydrodynamics is a macroscopic effective theory in which hydrodynamic modes are the relevant low energy degrees of freedom. In cases where the relevant degrees of freedom are actually prehydrodynamic modes, there is no guarantee that hydrodynamics or its simple generalizations will describe the system, just as hydrodynamics does not describe the physics of zero sound. This is not in contradiction to the recent result [16,19] that some non-trivial generalizations of hydrodynamics like an improved version of Israel-Stewart theory [20] and anisotropic and thirdorder hydrodynamics describe the bulk evolution of several simplified kinetic theory models even beginning at τ Redu . Rather, since these models include significant contributions from non-hydrodynamic modes, we emphasize that this observation alone does not imply that hydrodynamic modes dominate the evolution. It is worth exploring the applicability of these models in more general settings, however we hope that the identification of prehydrodynamic modes as a relevant slow degree of freedom may motivate the future construction of an effective theory of "pre-hydrodynamics". Identification of pre-hydrodynamic mode(s). -We consider a Bjorken-expanding medium of massless particles described by the kinetic equation where f (p z , p ⊥ ; τ ) is the single particle distribution, p ⊥ and p z are the transverse and longitudinal momentum, andĈ is the collision integral. Because of the symmetry, the only relevant hydrodynamic mode is the energy density . To more directly study the evolution of the energy density, we will focus on the momentum-weighted distribution function where p = p 2 ⊥ + p 2 z and θ = tan −1 (p z /p ⊥ ). Because the angular integration of F (cos θ; τ ) is the energy density, F (cos θ; τ ) describes the angular distribution of the particles that carry most of the energy. The p 3 -weighted moment of Eq. (3) gives the evolution equation for F : Following Ref. [21], we assume that f (p z , p ⊥ ; τ ) is symmetric under p z → −p z and expand F (cos θ; τ ) in a basis of the Legendre polynomials P 2n : Eq. (6) maps F (cos θ; τ ) to an infinite-dimensional vector ψ = ( , L 1 , L 2 , . . . , ). We therefore have the correspondence Note p L = 1 3 ( + 2L 1 ). Since F will become isotropic and approach in the hydrodynamic limit, the hydrodynamic mode corresponds to the vector The problem of hydrodynamization is therefore reduced to studying how ψ involves into φ H 0 . In the following discussion, we shall limit ourselves to the class of collision integrals for which Eq. (5) can be recast into the form where H is a non-Hermitian matrix and y = log(τ /τ I ) as we introduced earlier. This is satisfied for any collision integral that is linear in F (cos θ; τ ). Eq. (9) has the structure of the time-dependent Schrödinger equation in quantum mechanics. The explicit expression for the matrix H for RTA kinetic theory will be given in the subsequent section. Throughout this work, we will study the instantaneous eigenmodes φ n (y) of H(y). For clarity we will order them by the real part of their corresponding eigenvalues, e.g. ReE 0 < ReE 1 ≤ . . .. Of particular importance is the ground state mode φ 0 (y), which has the lowest damping rate of all of the eigenmodes. In the hydrodynamic limit τ ≥ τ Hydro , φ 0 (y) will evolve into φ H 0 since the conserved densities are the zero-modes of any collision kernel. At times τ < τ Hydro , we identify φ 0 (y) as the "pre-hydrodynamic mode" since it is an ancestor to the hydrodynamic mode φ H 0 . Emergent dominance of pre-hydrodynamic modes at τ τ C .-To illustrate the reduction in the degrees of freedom at early times, we study the behavior of ψ for τ τ C . In this case, ψ is determined by ∂ y ψ = −H F ψ where H F is obtained from Eq. (5) by neglecting the collision integral (the explicit expression can be obtained from the τ → 0 limit of Eq. (13)). To solve, we expand ψ in eigenstates of H F as ψ(τ ) = n=0 β n (τ )φ F n . It is easy to show that β n (τ ) = β n (τ I ) exp(−E n y) for all n. Therefore contributions from the "excited" modes φ F n>0 become unimportant after some emergent time scale The bulk evolution of the system around τ Redu is then dominated by the ground state mode φ F 0 . Related observations have also been made in Refs. [16,22].
For the description of heavy-ion collisions in the framework of perturbative QCD, τ I is of the order of Q −1 s , where Q s Λ QCD is the saturation scale (c.f. Refs. [23][24][25][26]). Meanwhile, a parametric estimate of τ C can be deduced from the collision integral, τ C Q s ∼ α −x s with exponent x > 0 (c.f. Ref. [12]). This hierarchy guarantees the existence of a time scale τ Redu that is parametrically smaller than τ Hydro ≥ τ C .
To appreciate the physics underlying the dominance of φ F 0 around τ Redu , we compare the explicit expression φ F 0 = (1, P 2 (0), P 4 (0) . . .) [18] with the definition in Eq. (6). It is then transparent that φ F 0 corresponds to an angular distribution function F (cos θ; τ ) that is sharply peaked at θ = π/2. For such a distribution, typical values of p z are much smaller than those of p ⊥ , meaning the longitudinal expansion drives arbitrary initial conditions to a highly anisotropic distribution in momentum space.
The analysis above shows that the longitudinal expansion together with the intrinsic hierarchy τ I τ C in weakly coupled QCD prepares the system in the instantaneous ground state φ F 0 . Since φ F 0 depends on H F but not on the initial conditions, the bulk evolution around τ Redu becomes insensitive to the details of the initial conditions. The latter has been observed in previous studies of kinetic theory [13,18,27], though its connection to the dominance of the mode φ F 0 has not been elucidated before. Implications of the dominance of prehydrodynamic modes. -We now explore the implications of the adiabatic evolution of H(y) after τ Redu . We begin by expanding ψ in terms of the instantaneous eigenmodes φ n (y) of H(y), ψ(y) = n=0 α n (y) φ n (y). While in general α n>0 can be the same order of magnitude as α 0 , under adiabatic evolution |α 0 | |α n>0 | and consequently ψ(y) ∼ φ 0 (y) .
Eq. (11) can be viewed as the definition of adiabatic hydrodynamization. We emphasize that this dominance of the prehydrodynamic mode φ 0 indicates that the bulk properties of the pre-equilibrium medium can be related to this mode and its eigenvalue. For example, let us focus on the percentage rate of change of the energy density in Eq. (1). Since the zeroth component of ψ is , it follows from Eqns. (9) and (11) that − g(y) is given by the zeroth component of Hψ, i.e. Eq. (2), even though g(y) in general can depend on all modes φ n (y). This non-trivial relation is a consequence of the adiabatic evolution. In the next section we will quantitatively test this result in the relaxation time approximation to determine the extent to which AH applies. RTA as an example of adiabatic hydrodynamization.- The collision integral under the relaxation-time approximation (RTA) iŝ where τ C is a function of y. Substituting Eqns. (12) and  2)). Black curves are g(λ) obtained from solving Eq. (3) with constant τC. After τ Redu , they collapse onto the RTA attractor as obtained in Refs. [7,10,13]. The left and right shaded regions indicate τ ≤ τ Redu and τ ≥ τ Hydro , respectively. (b) The fractional difference between E0 and g, which measures the relative importance of contributions from the ground state (pre-hydrodynamic) and excited modes. Red and dashed blue curves show this difference for representative initial conditions with constant and conformal τC, respectively. The fact that this quantity is small indicates the dominance of pre-hydrodynamic modes (i.e. adiabaticity) during the interval τ Redu ≤ τ < τ Hydro .

(6) into Eq. (5) gives [21]
where λ ≡ τ /τ C . Explicit expressions for a n , b n , and c n are given in Ref. [21], for example (a 0 , b 1 , c 0 ) = (4/3, 8/15, 2/3). From Eq. (13), the evolution of ψ has the form Eq. (9) with where the elements of H F , H 1 can be read from Eq. (13). From H RTA (λ) we compute the pre-hydrodynamic modes φ 0 (λ) and their energies E 0 (λ) for each λ. We note that the minimum gap ∆E min (λ) ≡ Re(E 1 (λ) − E 0 (λ)) is order one for λ 1 and becomes linear in λ for λ 1 1 . For all values of λ, φ 0 (λ) is gapped from the excited modes. It is easy to check that φ H 0 is the ground state of H 1 but not that of H RTA . Since H RTA evolves in time, the components of φ 0 (y) are different from those of φ H 0 for any finite y, exemplifying the distinction between prehydrodynamic and hydrodynamic modes.
The solid red curve in Fig. 1(a) shows E 0 (λ), which is the contribution to g(λ) from the pre-hydrodynamic modes only 2 . For comparison, we also determine g(τ ) by solving the kinetic equation numerically. Following Ref. [10], we use the parametrization τ C ∝ −∆/4 . Solutions to Eq. (3) with constant τ C (∆ = 0) and different initial conditions satisfying τ I τ C are shown in dashed black in Fig. 1(a). The resulting g collapses to a common curve at times much earlier than τ Hydro , which is the wellknown "attractor" behavior of Bjorken-expanding RTA kinetic theory. Remarkably, E 0 (λ) is close to g(τ ), indicating that the bulk evolution before the hydrodynamic regime is indeed dominated by the evolution of the prehydrodynamic modes.
Since the RTA attractor function g(τ ) has already been obtained by many authors [7,10,13], what is our purpose of studying this function? Our goal is to demonstrate that the main contribution to this function comes from pre-hydrodynamic modes. We emphasize that the attractor behavior of g(λ) alone does not tell us whether one mode or many modes are important for the subsequent evolution. In the language of quantum mechanics, the attractor behavior only indicates that the system is in its instantaneous ground state around τ Redu . The system remains in its instantaneous ground state here due to a qualitatively different reason, namely the suppression of transitions to the excited states.
To further demonstrate that the relative importance of contributions from excited modes are suppressed compared to those of the pre-hydrodynamic mode, we show the fractional difference δ ≡ |g − E 0 |/g as a function of λ in Fig. 1(b). Results for constant τ C (∆ = 0) and conformal τ C (∆ = 1) are shown in red and dashed blue, respectively. The evolution is more adiabatic the smaller δ is. δ is small both when λ 1 and λ 1 and reaches a maximum of 0.045 at intermediate λ. This indicates that at least 95% of the contribution to g between τ Redu and τ Hydro is from pre-hydrodynamic modes. We emphasize that δ is small even when the Knudsen number 1/λ is large. In fact, the contributions from the excited modes φ n>0 can be accounted for systematically by expanding in δ, generalizing the method developed in Refs. [11,28]. Including leading-order contributions from the excited states to g(λ) makes the adiabatic result in Fig. 1(a) essentially indistinguishable from the RTA attractor. This will be reported in upcoming work. Adiabaticity in the rapidly-expanding QGP.-Why does adiabaticity also apply to the violent expansion of the QGP in the early stages of the evolution? In essence, "adiabaticity" only requires that the transition to excited states is suppressed. For example, consider a time-dependent Hamiltonian in quantum mechanics H(t) = H 0 +λ(t)H 1 , where H 0 , H 1 are timeindependent andλ(t) is a monotonic function of time t. The transition rate from the instantaneous ground state |0, t to instantaneous excited states |n, t is given by ∂ t logλ/∆E n 0, t|λ(t)H 1 |n, t [11]. Therefore "adiabaticity" can arise either due to the smallness of the rate of change of the Hamiltonian ∂ t logλ compared to the energy gap ∆E (slow-quench adiabaticity), or due to the time-dependent part of the Hamiltonian 0, t|λH 1 |n, t being small in amplitude (fast-quench adiabaticity), see Ref [11] for examples of the applicability of adiabaticity to quantum phase transitions under fast quenches.
We have generalized the aforementioned quantum mechanical expression to a system described by Eq. (14). While the slow-quench adiabaticity applies at late times as one might expect, we also find that fast-quench adiabaticity applies at early times because λ is small. To see why this must be so on physical grounds, we recall that φ F 0 at very early times represents an angular distribution function F (cos θ; τ ) where typical values of p z are much smaller than those of p ⊥ . On the other hand, the excited states at early times have typical values of p z that are comparable to p ⊥ . A "transition" from the ground state to an excited state would therefore require either multiple scatterings or one rare hard scattering among gluons, the probability of which is suppressed when τ τ C .
Since our discussion above does not rely on the details of the collision integral, we expect that adiabaticity is a generic feature of both early-and late-time limits for the expanding weakly-coupled QGP. In particular, consider the standard bottom-up thermalization scenario [12]. Following the discussion above, we expect that adiabaticity applies during the stage τ Redu ≤ τ ≤ α −5/2 s Q −1 s and τ ≥ α −13/5 s Q −1 s . In the former stage, F (cos θ; τ ) represents the angular distribution of hard gluons (with typical energy Q s ) that rarely collide with one another. In the later stage, F (cos θ; τ ) represents the angular distribution of soft gluons (with typical energy T ) that are already in thermal equilibrium. Adiabaticity may break down during the transition stage α −5/2 ≤ Q s τ ≤ α −13/5 when the numbers of both soft and hard gluons are changing rapidly, however this interval is parametrically narrow compared to other stages.
Outlook.-While our analysis is based on a weakly-coupled kinetic description of the QGP, we anticipate that the concept of pre-hydrodynamic modes and the realization of AH is relevant more broadly. It would be interesting to explore AH for the QGP at strong coupling [29], and in table-top experiments [30]. As a first step towards this exploration, it may be necessary to develop a more general method to identify pre-hydrodynamic modes from the pole structure of off-equilibrium correlation functions.