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arXiv:2609.20705

Adiabatic hydrodynamization with transverse spatial gradients in boost-invariant plasmas

Uri Sharell, Jasmine Brewer, Weiyao Ke

Published Sep 17, 2026• 0 citations
arXivHTMLPDF

Abstract

The quantitative success of relativistic viscous hydrodynamics in describing the short-lived quark--gluon plasma raises a fundamental question: how does far-from-equilibrium QCD matter approach hydrodynamic behavior so rapidly? Microscopic kinetic-theory studies have related this onset to attractor dynamics, but typically assume transverse homogeneity. We relax this assumption by introducing gradient modes with finite transverse wave number $k$, extending the analysis of arXiv:2212.00820. These couple different spherical harmonics of the momentum distribution, and the resulting dynamics is controlled by the competition among the expansion rate $1/τ$, the collision rate $1/τ_R$, and the gradient scale $k$. At early times, longitudinal expansion suppresses this coupling, and different spherical harmonic sectors follow their homogeneous attractors. We find that, at later times and sufficiently small $k$, this gradient coupling drives the system toward a global attractor manifold spanned by hydrodynamic sound and shear modes, on a timescale $τ_D$ that depends on both the azimuthal harmonic $m$ and $kτ_R$. For sufficiently large $kτ_R$, the spectral gap closes: perturbations retain finite damping rates and therefore equilibrate, but no longer follow an isolated hydrodynamic attractor or admit a reduced description involving only a few hydrodynamic modes.

DOI
10.48550/arxiv.2609.20705

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The quantitative success of relativistic viscous hydrodynamics in describing the short-lived quark--gluon plasma raises a fundamental question: how does far-from-equilibrium QCD matter approach hydrodynamic behavior so rapidly? Microscopic kinetic-theory studies have related this onset to attractor dynamics, but typically assume transverse homogeneity. We relax this assumption by introducing gradient modes with finite transverse wave number kkk, extending the analysis of arXiv:2212.00820. These couple different spherical harmonics of the momentum distribution, and the resulting dynamics is controlled by the competition among the expansion rate 1/τ1/τ1/τ, the collision rate 1/τR1/τ_R1/τR​, and the gradient scale kkk. At early times, longitudinal expansion suppresses this coupling, and different spherical harmonic sectors follow their homogeneous attractors. We find that, at later times and sufficiently small kkk, this gradient coupling drives the system toward a global attractor manifold spanned by hydrodynamic sound and shear modes, on a timescale τDτ_DτD​ that depends on both the azimuthal harmonic mmm and kτRkτ_RkτR​. For sufficiently large kτRkτ_RkτR​, the spectral gap closes: perturbations retain finite damping rates and therefore equilibrate, but no longer follow an isolated hydrodynamic attractor or admit a reduced description involving only a few hydrodynamic modes.
The far-from equilibrium dynamics of the pre-hydrodynamic quarkgluon plasma (QGP) formed in heavy ion collisions can be characterized by distinct stages, during each of which the system loses some memory of its initial condition, until only the hydrodynamic modes remain. However, even though it has been repeatedly observed, finding intuitive physical explanations of how and why attractor behavior occurs has remained a challenge. The Adiabatic Hydrodynamization (AH) framework provides exactly such an explanation, showing that the attractor solution can be thought of as the ground state of an analog to quantum mechanical adiabatic evolution, provided we identify appropriate coordinate rescalings. Using the example of a simplified QCD kinetic theory in the small-angle scattering limit, we show how AH can explain both the early pre-hydrodynamic attractor and the later hydrodynamizing attractor in a longitudinally expanding gluon gas in a unified framework. By doing this, we provide a unified description of, and intuition for, all the stages of what in QCD would be bottom-up thermalization, starting from a pre-hydrodynamic attractor and ending with hydrodynamization.
Consistent formulations of relativistic viscous hydrodynamics involve short-lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Müller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.
The far-from equilibrium dynamics of the pre-hydrodynamic quarkgluon plasma (QGP) formed in heavy ion collisions can be characterized by distinct stages, during each of which the system loses some memory of its initial condition, until only the hydrodynamic modes remain. However, even though it has been repeatedly observed, finding intuitive physical explanations of how and why attractor behavior occurs has remained a challenge. The Adiabatic Hydrodynamization (AH) framework provides exactly such an explanation, showing that the attractor solution can be thought of as the ground state of an analog to quantum mechanical adiabatic evolution, provided we identify appropriate coordinate rescalings. Using the example of a simplified QCD kinetic theory in the small-angle scattering limit, we show how AH can explain both the early pre-hydrodynamic attractor and the later hydrodynamizing attractor in a longitudinally expanding gluon gas in a unified framework. By doing this, we provide a unified description of, and intuition for, all the stages of what in QCD would be bottom-up thermalization, starting from a pre-hydrodynamic attractor and ending with hydrodynamization.
Attractors are effective low-dimensional structures, defined in a chosen set of observables, that trajectories from different initial states approach; hydrodynamic attractors are those on which the late-time evolution is governed by hydrodynamic constitutive relations. Motivated by nuclear collisions and ultracold atomic gases, this chapter distinguishes attractorization (the loss of sensitivity to some directions in the space of initial states) from hydrodynamization (the onset of validity of hydrodynamic constitutive relations). Using mainly conformal, boost-invariant Bjorken flow, we compare Müller-Israel-Stewart-type theories, kinetic theory, holography, and classical Yang-Mills fields. Forward attraction, produced by the decay of non-hydrodynamic modes, is distinguished from pullback attraction, which can select a unique regular solution, and from expansion-driven attraction, which can suppress initial-state sensitivity before microscopic relaxation, with or without a subsequent fluid regime. Divergent gradient expansions and their transseries completions, the state-space picture, and adiabatic hydrodynamization provide complementary descriptions. Applications to nuclear collisions include particle production, transverse energy and flow, the initialization of and linear response around attracting backgrounds, jet quenching, and attractor-informed modifications of hydrodynamic models. For ultracold Fermi gases, we review theoretical proposals for bulk-channel attractors under scattering-length drives and distinguish them from recent measurements of short-time contact and momentum-distribution dynamics.
Nonthermal attractors govern the emergent self-similar dynamics of far-from-equilibrium quantum systems, from ultrarelativistic nuclear collisions to cold-atom experiments. Within the framework of adiabatic hydrodynamization, the approach to a nonthermal attractor is described by the decay of excited states of an effective Hamiltonian. Using an exactly solvable kinetic theory-the longitudinally expanding, overoccupied gluon plasma dominated by small-angle elastic scattering-we establish a direct correspondence between the eigenmodes of adiabatic hydrodynamization and the quasinormal mode spectrum of the nonthermal attractor. This equivalence suggests a general framework for identifying universal dynamical structures in nonequilibrium systems. As a byproduct, we derive analytic prescaling solutions for strongly longitudinally expanding systems.
Nonequilibrium Green's functions provide an efficient way to describe the evolution of the energy-momentum tensor during the early-time preequilibrium stage of high-energy heavy-ion collisions. Besides their practical relevance they also provide a meaningful way to address the question when and to what extent a hydrodynamic description of the system becomes applicable. Within the kinetic theory framework we derive a new method to calculate time-dependent nonequilibrium Green's functions describing the evolution of energy and momentum perturbations on top of an evolving far-from-equilibrium background. We discuss the approach toward viscous hydrodynamics along with the emergence of various scaling phenomena for conformal systems. By comparing our results obtained in the relaxation time approximation to previous calculations in Yang-Mills kinetic theory, we further address the question which macroscopic features of the energy-momentum tensor are sensitive to the underlying microscopic dynamics.