From Bjorken Flow to Ultracold Gases: The Historical Development of Hydrodynamic Attractors

From Bjorken flow to the attractor concept: what earlier work established

The hydrodynamic attractor concept emerged from attempts to understand why viscous hydrodynamics describes quark-gluon plasma in nuclear collisions even when the system is small, rapidly expanding, and far from local equilibrium [1]. Bjorken flow—longitudinally boost-invariant expansion with transverse homogeneity—reduces the dynamics to proper time, and the dimensionless clock w = τT measures expansion strength [1]. In a conformal Müller-Israel-Stewart-type model with the nonlinear second-order structure of BRSSS theory, solutions with different initial dissipative stresses were found to approach a common relation between pressure anisotropy and w [1]. This relation reproduces the hydrodynamic gradient expansion at large w but remains predictive at smaller w where low-order truncations fail [1]. Romatschke generalized this to resummed BRSSS theory, kinetic theory in the relaxation time approximation, and strongly coupled N=4 super Yang-Mills theory, showing that arbitrary initial data quickly approach attractor solutions via nonhydrodynamic mode decay [6]. The gradient series was found to diverge but be Borel summable, giving rise to a nonanalytic attractor well defined even for large gradients [6].

Attractorization is not hydrodynamization: the review's central distinction

The review's central conceptual contribution is distinguishing attractorization—reduced sensitivity to some directions in the space of initial conditions—from hydrodynamization, the onset of validity of hydrodynamic constitutive relations [1]. The attracting set need not be a single curve, and energy scales or conserved densities can remain as coordinates along it [1]. Different moments of a kinetic distribution can approach their attracting behavior at different rates [1]. The review further separates three mechanisms: forward attraction, produced by the decay of nonhydrodynamic modes at late times; pullback attraction, which keeps observation time fixed while moving initialization earlier toward the singular early-time endpoint, selecting a unique regular solution in conformal BRSSS theory; and expansion-driven attraction, which can suppress initial-state sensitivity algebraically before microscopic relaxation becomes effective [1]. This last mechanism can precede hydrodynamization, retain information in other observables, or occur without a subsequent fluid regime [1]. The reduced classical Yang-Mills example demonstrates the possibility of partial recovery of previously suppressed sensitivity [1].

Exact attractors and convergent expansions: a counterexample to universal divergence

Denicol and Noronha derived the general analytical solution of viscous hydrodynamic equations for an ultrarelativistic gas of hard spheres undergoing Bjorken expansion, including particle number conservation, and analytically determined its late-time attractor [5]. Differently from all previous cases involving rapidly expanding fluids, the gradient expansion converges absolutely in a finite range of Knudsen numbers [5]. They exactly determined the hydrodynamic attractor when microscopic dynamics is modeled by the Boltzmann equation with a fully nonlinear collision kernel [5]. The Israel-Stewart attractor and the full Boltzmann attractor agree qualitatively, with visible quantitative deviations only at large Knudsen numbers, remaining at most 20% [5]. This shows that the divergence of the gradient expansion is not a general property of rapidly expanding systems nor a singular feature of Bjorken flow [5]. The review contextualizes this by noting that in holographic Bjorken flow, conformal BRSSS theory, and relaxation-time kinetic theory, the late-time gradient series exhibits factorial growth, but this statement concerns those expansions, not every series used in hydrodynamics [1].

Cyclic attractors in periodically driven ultracold Fermi gases

Mazeliauskas and Enss demonstrated that a system undergoing periodic expansion and contraction exhibits a novel cyclic attractor behavior [2][4]. They employed Müller-Israel-Stewart theory to a driven ultracold Fermi gas, predicting the shape of the attractor, which does not converge to Navier-Stokes dynamics at late times [2][4]. The protocol varies the scattering length a(t) periodically via Feshbach resonances, which is equivalent to fluid expansion and probes local dissipation even in a uniform system without actual fluid motion [2][4]. In linear response, the bulk pressure follows the MIS equation τ_ζ Π̇(t) = −Π(t) + Π_NS(t), where τ_ζ is the relaxation time [2][4]. For faster drive, plotting the full bulk pressure parametrically against the Navier-Stokes prediction reveals an elliptical trajectory; trajectories for different initial equilibrium states converge toward the same ellipse, identified as a cyclic attractor curve [1]. This cyclic attractor is conceptually distinct from monotonic attractors because it does not converge to Navier-Stokes at late times and therefore cannot be described through late-time constitutive behavior [1]. For larger-amplitude drives, nonlinear response was modeled in massive relativistic RTA kinetic theory, showing convergence toward an attractor curve that no longer closes because the system heats up under the drive [1].

Early-time attractors, expansion-driven contraction, and few-body scaling

The review discusses a recent proposal studying universality emergence through strong expansion at short times, an instance of expansion-driven attraction [1]. Tracking joint evolution of bulk pressure and energy density, solutions for sets of initial conditions contract along one direction in state space toward a functional relation between Π(t) and ε(t) already at short times t ≲ τ_Π [1]. In contrast to Bjorken flow, the form of this functional relation evolves with time and shows no early-time limiting fixed point [1]. However, the contraction demonstrates that expansion can provide a mechanism of information loss besides equilibration in cold atoms, allowing experimental verification [1]. The review also distinguishes short-time few-body scaling from many-body hydrodynamic attraction [1]. After a sudden increase of interaction strength, the pairing amplitude grows and initially converges onto a few-body conformal attractor η ∼ (D₀t)^{1/2} before relevant perturbations lead away [1]. Momentum distribution change measured in ultracold fermionic atoms after a sudden change of interaction has been compared to scaling predictions [1]. The review emphasizes that these rapid contact measurements, contact dynamics in a Bose gas, and momentum-distribution dynamics in quenched Fermi gases provide complementary experimental advances, not interchangeable demonstrations of the theoretical attractor constructions [1].

Adiabatic hydrodynamization and the quasinormal mode connection

De Lescluze, Heller, Mazeliauskas, Scheihing-Hitschfeld, and Werthmann established a direct correspondence between the eigenmodes of adiabatic hydrodynamization and the quasinormal mode spectrum of the nonthermal attractor [3]. Using an exactly solvable kinetic theory—the longitudinally expanding, overoccupied gluon plasma dominated by small-angle elastic scattering—they showed this equivalence [3]. Within adiabatic hydrodynamization, the approach to a nonthermal attractor is described by the decay of excited states of an effective Hamiltonian [3]. The review notes that adiabatic hydrodynamization offers a possible common framework: in adapted variables, rapidly damped modes become subdominant to a slowly evolving sector, provided spectral separation is accompanied by sufficiently weak mixing between the sectors [1]. Scaling can simplify this construction but is neither necessary nor sufficient for attraction [1]. The explicit connection between this framework and quasinormal modes provides a concrete bridge between perspectives [1]. As a byproduct, the authors derived analytic prescaling solutions for strongly longitudinally expanding systems [3].

Nuclear collision applications and the limits of attractor-informed modeling

The review details phenomenological implications for nuclear collisions, including particle production, transverse energy and flow, initialization of and linear response around attracting backgrounds, jet quenching, and attractor-informed modifications of hydrodynamic models [1]. The energy-density attractor connects early energy deposition to subsequent cooling and entropy production, yielding estimates of final multiplicity under stated assumptions [1]. Applied locally, it constrains transverse energy, evolving geometry, and initial buildup of transverse flow [1]. These results inform consistent hydrodynamic initialization and linear response of transverse perturbations around an attracting background, as implemented in KøMPøST [1]. For jet quenching, the review notes that effects in the Glasma and kinetic theory stages were found to be sizable, but event-by-event descriptions typically cannot use these more expensive descriptions and instead use formulae derived for a homogeneous isotropic medium in equilibrium, extrapolated beyond this strict limit [1]. The review also discusses perturbations of global Bjorken flow, where the solution for spatially inhomogeneous perturbations can be written as a transseries with the background attractor stable against these perturbations, and all transverse plane information is encoded in coefficients of this transseries expansion [1]. However, the accuracy of the linearization scheme, which may be a rough approximation especially in the outskirts of the system, has not been tested [1].

Open questions and the boundary between theory and measurement

The review states that attraction is relative to specified observables and initial states; its mechanism and onset need not coincide with hydrodynamization; and its phenomenological usefulness does not establish a universal constitutive theory beyond the flow and response that have been tested [1]. The main protocols for ultracold gases vary the scattering length of a spatially uniform, normal Fermi gas near unitarity and probe the bulk channel, absent in exactly conformal Bjorken hydrodynamics [1]. Monotonic driving, periodic driving, and early expansion-driven contraction test different dynamical regimes, but these remain theoretical attractor constructions [1]. Rapid contact measurements in Fermi gases, contact dynamics in a Bose gas, and momentum-distribution dynamics in quenched Fermi gases provide complementary experimental advances, not interchangeable demonstrations of those constructions [1]. In particular, short-time few-body scaling must be distinguished from many-body hydrodynamic attraction [1]. The review notes that holographic hydrodynamization commonly occurs at w of order unity, while kinetic pre-equilibrium calculations can support matching to hydrodynamics around 1 fm/c for phenomenologically motivated parameters; neither statement defines a universal hydrodynamization time [1]. An all-order relation in a symmetric flow is not a constitutive law for arbitrary space-time dependence, and distinct gradient structures become indistinguishable or vanish under Bjorken symmetry [1]. Moreover, higher-order coefficients generated by a finite-parameter MIS-type model are predictions of that completion, not independently matched microscopic transport data [1].

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This research page is built on 6 studies (5 peer-reviewed, 1 preprint) — published from 2018 to 2026, 4 from 2024 or later — selected as the most relevant from 10 studies that passed quality screening, drawn from 60 papers retrieved from a database of over 500 million.

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