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arXiv:2609.09114

Hydrodynamic attractors

Tilman Enss, Michał P. Heller, Alexandre Serantes, Clemens Werthmann

Published Sep 8, 2026• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2609.09114

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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.

Hydrodynamic attractors characterize hydrodynamiclike evolution in strongly interacting systems, independent of initial conditions or microscopic details, outside the conventional hydrodynamic regime. They explain why hydrodynamic models apply to high-energy nuclear collisions, but so far have only been explored for monotonic expansion, such as Bjorken flow. We demonstrate that a system undergoing periodic expansion and contraction exhibits a novel cyclic attractor behavior. We employ Müller-Israel-Stewart theory to a driven ultracold Fermi gas to predict the shape of the attractor, which does not converge to Navier-Stokes dynamics at late times. This phenomenon can be measured in real time in ultracold quantum gases with externally modulated scattering length, offering a new avenue for the experimental discovery of hydrodynamic attractors.

The hydrodynamic attractor is a concept that describes universal equilibration behavior in which systems lose microscopic details before hydrodynamics becomes applicable. We propose a setup to observe hydrodynamic attractors in ultracold atomic gases, taking advantage of the fact that driving the two-body s-wave scattering length causes phenomena equivalent to isotropic fluid expansions. We specifically consider two-component fermions with contact interactions in three dimensions and discuss their dynamics under a power-law drive of the scattering length in a uniform system. By explicit computation, we derive a hydrodynamic relaxation model. We analytically solve their dynamics and find the hydrodynamic attractor solution. Our proposed method using the scattering length drive is applicable to a wide range of ultracold atomic systems, and our results establish these as a new platform for exploring hydrodynamic attractors.

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.

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.

Hydrodynamics is a macroscopic theory of long-wavelength dynamics around local thermal equilibrium. We develop attractodynamics, the analogous construction around a far-from-equilibrium attractor. Dynamics near a far-from-equilibrium attractor retains some non-hydrodynamic microscopic information, which attractodynamics systematically organizes. We move towards this general structure by starting in 0+1D, and consider a model for which an anisotropic far-from-equilibrium attractor solution is exactly known. This setting provides a clean benchmark in which the ideal attractodynamic equations and a leading transient residual extension can be compared directly with the full kinetic evolution. The resulting hierarchy gives an improvable description of near-attractor dynamics: the ideal theory captures evolution close to the attracting manifold, while retaining the leading off-attractor moments extends the regime of agreement until the finite truncation breaks down. This example identifies the ingredients needed for local 3+1D attractodynamics and for macroscopic attractodynamic theories not derived from an underlying kinetic description.

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.