Holographic Hydrodynamization, Attractors, and Gradient Asymptotics

Hydrodynamization means that a stress tensor is accurately described by a hydrodynamic constitutive relation; it does not require pressure isotropy or local equilibrium. In boost-invariant holographic flow, many initial geometries approach a common curve in suitable dimensionless variables, while exponentially decaying quasinormal sectors carry their residual initial-state dependence. The gradient series is asymptotic, so the attractor requires a resummation prescription and an uncertainty estimate.

Required background. Bulk Quasinormal Modes and Boundary Hydrodynamic Poles supplies transient modes; Time-Dependent Geometries and Holographic Thermalization supplies gravitational evolution; Hydrodynamic Attractors and Asymptotic Gradient Expansions fixes the QFT definition.

Helpful background. Hydrodynamization and the Kinetic-to-Hydrodynamic Map provides a nonholographic comparison.

Bjorken-flow variables

For boost-invariant, transversely homogeneous flow,

$$T^\mu{}_\nu =\operatorname{diag} \left(\epsilon(\tau),-p_L(\tau),-p_T(\tau),-p_T(\tau)\right),$$

and conservation plus conformality leave one independent function. Define

$$w=\tau T_{\mathrm{eff}}(\tau), \qquad \mathcal A(w)=\frac{p_T-p_L}{\epsilon}.$$

Hydrodynamics predicts an asymptotic series

$$\mathcal A_{\mathrm{hydro}}(w) \sim\sum_{n=1}^{\infty}\frac{a_n}{w^n}.$$

In holographic $\mathcal N=4$ plasma the coefficients grow factorially, so the series has zero radius of convergence Heller, Janik, and Witaszczyk 2013. Its Borel singularities are related to nonhydrodynamic sectors.

Transients and the first application

Solve the asymptotically AdS Einstein equations for several regular initial radial profiles with the same energy scale. Holographic renormalization gives $\epsilon(\tau)$ and hence $\mathcal A(w)$. Compare with a truncated or Borel-resummed hydrodynamic curve using

$$\delta_{\mathrm{hydro}}(w) =\frac{\lvert\mathcal A_{\mathrm{num}}(w) -\mathcal A_{\mathrm{resum}}(w)\rvert} {\max(1,\lvert\mathcal A_{\mathrm{num}}(w)\rvert)}.$$

The leading residual has a transseries form

$$\delta\mathcal A \sim \sigma\,w^\beta e^{-\Omega w} \left[1+O(w^{-1})\right],$$

where $\Omega$ is tied to the least damped relevant quasinormal frequency and $\sigma$ depends on initial data. Hydrodynamization occurs when $\delta_{\mathrm{hydro}}$ falls below a declared tolerance, even if $p_L/p_T$ remains far from one. Early numerical demonstrations of this separation were given by Heller, Janik, and Witaszczyk 2012.

Initial-state and resummation adversary

Change the regular bulk initial profile while keeping the same final scale. If the curves approach one another only after a redefinition that depends on the initial condition, the claimed universal attractor has been partly calibrated rather than predicted.

Next change the Borel contour or transseries completion. The spread between admissible resummations estimates an ambiguity that the finite gradient coefficients cannot remove. A stable attractor claim requires the numerical solutions to lie within this spread and the inferred hydrodynamization time to remain stable under resolution, initial-profile, and tolerance changes. Resurgent hydrodynamic completion was formulated explicitly by Heller and Spaliński 2015.

Evidence ceiling and handoff

The calculation establishes an attractor and transient spectrum for the specified holographic theory, symmetry class, initial-data family, normalization, and error metric. It does not identify the initial state of a real collision or make the same hydrodynamization time universal across kinetic theory and QCD.

Thermal and Nonequilibrium QFT owns attractor theory; Chapter 11 owns phenomenological fluid applications. The classical saddle does not determine exact finite-$N$ late time.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Heller, Michał P.; Janik, Romuald A.; and Witaszczyk, Przemysław. “Characteristics of Thermalization of Boost-Invariant Plasma from Holography.” Physical Review Letters 108, 201602 (2012). doi:10.1103/PhysRevLett.108.201602.
• Heller, Michał P.; Janik, Romuald A.; and Witaszczyk, Przemysław. “Hydrodynamic Gradient Expansion in Gauge Theory Plasmas.” Physical Review Letters 110, 211602 (2013). doi:10.1103/PhysRevLett.110.211602.
• Heller, Michał P., and Michał Spaliński. “Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation.” Physical Review Letters 115, 072501 (2015). doi:10.1103/PhysRevLett.115.072501.