Adiabatic Hydrodynamization in QGP

• Adiabatic Hydrodynamization (AH) is a framework that describes how the quark–gluon plasma transitions from an initial far-from-equilibrium state to a fluid governed by hydrodynamics through adiabatic evolution of slow modes.
• It uses kinetic theory and Bjorken expansion to show that rapid longitudinal flow reduces active degrees of freedom, leading to a hydrodynamic attractor that governs energy density and pressure anisotropy.
• The methodology connects early pre-hydrodynamic anisotropies with viscous hydrodynamic modes, offering quantitative insights into hydrodynamization times and the evolution of pressure ratios in heavy-ion collisions.

Adiabatic Hydrodynamization (AH) is a microscopic scenario for the transition of the quark–gluon plasma produced in heavy-ion collisions from a far-from-equilibrium state to a fluid governed by hydrodynamics. Its central claim is that the bulk evolution is controlled, already during the pre-equilibrium stage, by a small set of slow degrees of freedom that are not yet hydrodynamic modes but evolve continuously into them as interactions isotropize momentum distributions and conserved densities take over. In this formulation, rapid Bjorken expansion dynamically reduces the effective number of active degrees of freedom, and the resulting slow-mode dominance provides a kinetic-theory explanation of the hydrodynamic attractor and of the near-universal evolution of energy density and pressure anisotropy in boost-invariant flow (Brewer et al., 2019).

1. Conceptual structure and physical meaning

AH is formulated by analogy with adiabatic evolution in quantum mechanics, but for a non-Hermitian, time-dependent kinetic generator. The state of the system is decomposed into instantaneous eigenmodes of an operator $H(\tau)$, and the relevant mode at each time is the right eigenvector with the smallest damping rate. This mode is the “instantaneous ground state,” while higher modes are separated by spectral gaps and decay more rapidly. Early in the evolution this ground state is “pre-hydrodynamic”: it is not associated with conserved densities and does not yet have the form of a hydrodynamic mode. As collisions accumulate, however, it morphs continuously into the hydrodynamic mode associated with energy conservation (Brewer et al., 2019).

The physical driver of this reduction is rapid longitudinal expansion. In Bjorken flow, the longitudinal momentum redshifts as $p_z \sim p_z(\tau_0)\tau_0/\tau$ while $p_T$ remains unchanged, so the system rapidly develops $p_z \ll p_T$. This kinematic sieve suppresses excited angular harmonics and concentrates the energy-carrying distribution near $\theta=\pi/2$. AH therefore does not identify hydrodynamization with equilibration or isotropization. In the holographic language, hydrodynamization means that the one-point function $T^{\mu\nu}$ becomes accurately describable by hydrodynamic constitutive relations and conservation laws even while the state remains far from local equilibrium and may retain large pressure anisotropy (Heller, 2016).

2. Kinetic-theory formulation in Bjorken flow

The basic weak-coupling setting is a conformal, boost-invariant system of massless particles obeying the Boltzmann equation in Milne time,

$$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$

In the relaxation-time approximation, the competition between expansion and collisions is summarized by

$w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$

which measures the relative importance of interactions and expansion. The energy-momentum tensor is

$$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$

with $p^0=|\mathbf p|$ for massless particles. Under Bjorken symmetry the relevant observables are the energy density $p_z \sim p_z(\tau_0)\tau_0/\tau$0, the longitudinal pressure $p_z \sim p_z(\tau_0)\tau_0/\tau$1, and the transverse pressure $p_z \sim p_z(\tau_0)\tau_0/\tau$2 (Brewer et al., 2019).

A particularly useful object is the $p_z \sim p_z(\tau_0)\tau_0/\tau$3-weighted angular distribution

$p_z \sim p_z(\tau_0)\tau_0/\tau$4

whose angular average is $p_z \sim p_z(\tau_0)\tau_0/\tau$5. Expanding in even Legendre harmonics,

$p_z \sim p_z(\tau_0)\tau_0/\tau$6

one finds

$p_z \sim p_z(\tau_0)\tau_0/\tau$7

The coefficient $p_z \sim p_z(\tau_0)\tau_0/\tau$8, corresponding to the quadrupole $p_z \sim p_z(\tau_0)\tau_0/\tau$9 harmonic, encodes the leading anisotropy that drives $p_T$0 away from unity. In AH this quadrupole is identified as the dominant pre-hydrodynamic slow mode of the energy-carrying gluons (Brewer et al., 2019).

The moment hierarchy can be organized as

$p_T$1

where $p_T$2 and $p_T$3 is a non-Hermitian generator encoding expansion and collisions. The instantaneous right-eigenmodes of $p_T$4 provide the natural basis for AH.

3. Adiabatic following, coordinate rescalings, and prehydrodynamic attractors

The adiabatic statement is that the evolving state remains predominantly aligned with the instantaneous ground state, with only small admixtures from excited modes. In the RTA Bjorken setting, a practical decomposition is

$p_T$5

and the generalized adiabaticity condition is

$p_T$6

Two sufficient regimes were identified. In the slow-quench regime, $p_T$7 is large and varies slowly, so spectral gaps grow with $p_T$8. In the fast-quench regime, $p_T$9, the collisional piece is weak, and transitions to excited states require multiple or hard scatterings that are rare compared with expansion. AH is therefore possible both when the system is interaction-dominated and when it is still far from equilibrium but expansion-dominated (Brewer et al., 2019).

Later work generalized this idea into an explicitly model-independent coordinate-construction procedure. One writes

$p_z \ll p_T$0

with scaling exponents

$p_z \ll p_T$1

and adiabatic time $p_z \ll p_T$2. In these variables the kinetic equation becomes a pseudo-Schrödinger problem,

$p_z \ll p_T$3

The AH prescription chooses $p_z \ll p_T$4 and $p_z \ll p_T$5 so as to minimize the time variation of the ground state of $p_z \ll p_T$6, and chooses $p_z \ll p_T$7 so that the ground-state eigenvalue vanishes, $p_z \ll p_T$8. The attractor in the transformed space is then the instantaneous ground-state manifold of $p_z \ll p_T$9 (Rajagopal et al., 2023).

In the small-angle kinetic theory of a longitudinally expanding gluon gas, this formulation reproduces the familiar BMSS early-time scaling form

$\theta=\pi/2$0

with $\theta=\pi/2$1, and also organizes the later dilute and hydrodynamic regimes. In the unified description, the characteristic scaling exponents are $\theta=\pi/2$2 for free streaming, $\theta=\pi/2$3 for BMSS, $\theta=\pi/2$4 for the dilute fixed point, and $\theta=\pi/2$5 for late hydrodynamic scaling (Rajagopal et al., 2024).

4. From pre-hydrodynamic modes to hydrodynamic modes

A central AH result is that the dominant early-time quadrupole anisotropy is not discarded when hydrodynamics becomes accurate; rather, it evolves continuously into the hydrodynamic shear mode. In viscous hydrodynamics, the leading correction to local equilibrium has the standard quadrupolar form

$\theta=\pi/2$6

so the angular structure of the pre-hydrodynamic mode matches the angular structure of the late-time viscous correction. This provides a microscopic account of why a single angular harmonic can control both the far-from-equilibrium stage and the viscous stage (Brewer et al., 2019).

At strong coupling, the same physical theme appears in a different language. Holography describes transient, nonhydrodynamic excitations as quasinormal modes of the dual AdS black brane, with damping rates $\theta=\pi/2$7. AH in this setting means that the macroscopic rate of change becomes small compared with those damping rates. A useful criterion is

$\theta=\pi/2$8

For Bjorken flow, with $\theta=\pi/2$9, this becomes the statement that hydrodynamization occurs for $T^{\mu\nu}$0. The same framework also clarifies why the hydrodynamic gradient expansion is asymptotic: the large-order behavior is controlled by nonhydrodynamic QNMs, and the Borel-plane singularities track the least-damped QNM frequencies (Heller, 2016).

An exactly solvable small-angle model sharpened the connection further by mapping AH excited states to the quasinormal modes of a nonthermal attractor. In that formulation, the rescaled evolution operator has time-independent Hermite-Gaussian eigenfunctions, the ground state is the scaling solution, and the excited-state amplitudes decay as powers of the rescaling factor $T^{\mu\nu}$1. The QNM spectrum includes a “prescaling QNM” with $T^{\mu\nu}$2 and higher modes with $T^{\mu\nu}$3 for $T^{\mu\nu}$4, making the AH–QNM correspondence explicit in a far-from-equilibrium kinetic setting (Lescluze et al., 16 Oct 2025).

5. Quantitative behavior, universality, and attractors without scaling

In RTA Bjorken kinetics, the quantity

$T^{\mu\nu}$5

is approximately given by the instantaneous ground-state eigenvalue,

$T^{\mu\nu}$6

After truncation of the harmonic hierarchy to $T^{\mu\nu}$7, a variety of initial conditions and relaxation-time profiles collapse onto the same attractor well before $T^{\mu\nu}$8. The deviation

$T^{\mu\nu}$9

remains small throughout the pre-hydrodynamic stage; specifically, it is $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$0 across $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$1, so the dominant pre-hydrodynamic mode supplies more than $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$2 of the evolution of $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$3. For RHIC/LHC-like conditions with $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$4–$$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$5 GeV and $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$6–$$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$7, one finds $$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$8–$$\partial_\tau f(p_T,p_z;\tau)-\frac{p_z}{\tau}\,\partial_{p_z}f(p_T,p_z;\tau)=C[f].$$9 fm/c, implying that $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$0 passes unity by $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$1–$w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$2 fm/c (Brewer et al., 2019).

The later kinetic-theory literature emphasized that AH does not require exact prethermal scaling. In boost-invariant gluon kinetics with and without number non-conserving inelastic processes, one finds a band of low-effective-energy modes that evolves adiabatically long before hydrodynamization; the emergence of a gap between that ground-state band and the excited modes coincides with the fall onto a prehydrodynamic attractor surface, even when no approximate scaling exponents $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$3 are visible. First-order hydrodynamics becomes applicable at the same time that a unique ground state emerges. In the simplified model studied there, inelastic scattering reduces the hydrodynamization time from $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$4–$w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$5 to $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$6 at $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$7, while at $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$8 one finds $w \equiv \frac{\tau}{\tau_R(\tau)}=\lambda,$9 (Rajagopal et al., 28 Jul 2025).

Strong-coupling calculations provide complementary benchmarks. In holographic Bjorken flow, widely different initial conditions collapse onto hydrodynamic behavior already at $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$0–$$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$1, and a survey of 600 initial states gave an average hydrodynamization time $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$2. At that point the pressure anisotropy remains large: one estimate gives $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$3, corresponding to $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$4 (Heller, 2016, Jankowski, 2015).

6. Scope, limitations, and related uses of the term

AH is often contrasted with other pre-equilibrium frameworks. Free-streaming models capture the generation of early anisotropy but not the continuous adiabatic following into hydrodynamics. Anisotropic hydrodynamics assumes that hydrodynamic variables remain the relevant slow modes far from equilibrium, whereas AH shows that in Bjorken RTA the early slow mode is qualitatively different from the hydrodynamic mode and becomes hydrodynamic only through continuous evolution. This distinction is central to the AH interpretation of attractors and early memory loss (Brewer et al., 2019).

The current quantitative results remain model-dependent. Much of the explicit evidence relies on weak coupling, conformal symmetry, Bjorken expansion, and either RTA or simplified small-angle elastic and inelastic kernels. Realistic QCD introduces number-changing processes, Landau–Pomeranchuk–Migdal suppression, running coupling, quarks, transverse expansion, and nonconformal effects. Work with simplified $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$5 kernels indicates that including number non-conserving processes restores several qualitative features of QCD effective kinetic theory and yields more realistic hydrodynamization times, but it does not eliminate the need for more complete collision integrals (Rajagopal et al., 2024, Rajagopal et al., 28 Jul 2025).

A separate terminological issue is that “adiabaticity” also has an established meaning in the transport-classification literature. There, “adiabatic fluids” satisfy an off-shell adiabaticity equation in which the entropy-current divergence is compensated by $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$6- and $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$7-weighted Ward identities, leading to the “eightfold way” classification of transport and to the proposal of an emergent $$T^{\mu\nu}(\tau)=g\int \frac{d^3p}{(2\pi)^3\,p^0}\,p^\mu p^\nu f(\tau,\mathbf p),$$8 or macroscopic KMS symmetry. That notion concerns the taxonomy of hydrodynamic transport and is distinct from the later AH framework for hydrodynamization in expanding plasmas (Haehl et al., 2015).

This suggests that the AH intuition of timescale separation, slow manifolds, and rapid loss of memory may have broader relevance beyond heavy-ion kinetics. A cold-atom experiment directly observed hydrodynamization preceding local prethermalization in an array of 1D Bose gases, with the fast hydrodynamization timescale set by the Bragg energy scale and the slower local prethermalization time scaling inversely with momentum. Although that work did not formulate the dynamics in the AH spectral language, it provided a direct example of the separation between rapid hydrodynamization and later local equilibration that AH was designed to explain (Le et al., 2022).