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This chapter explains a simple fact that can seem surprising: systems far from equilibrium often evolve in ways that look like fluid motion sooner than one might expect. The authors review the idea of a hydrodynamic attractor. In plain terms, an attractor is a low-dimensional relation between a few observables that many different initial states converge to. A hydrodynamic attractor is one where that late-time relation matches the rules (constitutive relations) used in viscous fluid dynamics.
The chapter compares several theoretical approaches to show how attractors appear. The main testing ground is Bjorken flow, a highly symmetric model of longitudinal expansion used in studies of nuclear collisions. The authors discuss Müller–Israel–Stewart (MIS) type relaxation models and the BRSSS second-order theory (ways to write viscous fluid equations with relaxation), kinetic theory in the relaxation-time approximation (RTA), holography (a strong‑coupling calculational tool that maps some quantum systems to gravity problems), and classical Yang–Mills field simulations. Across these frameworks, different non-hydrodynamic excitations die away and many trajectories move toward a common relation between pressure anisotropy and a dimensionless measure of expansion.
The review also clarifies two different concepts that are often mixed up. Attractorization means that the system loses sensitivity to some directions in the space of initial states, so many different starts end up near the same curve. Hydrodynamization means the fluid constitutive relations become a good approximation to the true stress and currents. These can happen on different timescales. For example, rapid expansion can suppress initial differences before microscopic collisions have time to relax the system (called expansion-driven attraction). That suppression does not always imply the system has become a true fluid.
At the technical level, the chapter explains why attractors are connected to the failure of simple gradient expansions. In several cases the derivative expansion of hydrodynamics grows factorially at large order, so it does not converge. Resummation techniques introduce a transseries: the usual gradient terms plus extra nonperturbative sectors that are exponentially small at late times. Those extra pieces carry information about initial conditions. Resumming the series and cancelling ambiguities does not by itself fix the amplitudes of these sectors, so some memory of the initial state can survive.
Why this matters: hydrodynamic attractors help explain why viscous fluid models work well for the quark–gluon plasma created in high-energy nuclear collisions, even when pressures are very anisotropic and the system is small and rapidly expanding. The chapter lists concrete applications to particle production, transverse energy and flow, how to initialize hydrodynamic simulations, linear response around attracting backgrounds, and jet quenching. It also points to proposals for seeing related attractor behavior in ultracold Fermi gases when the interaction strength is driven in time.
Important caveats are emphasized. Results found in the symmetric Bjorken flow setup do not automatically generalize to arbitrary space‑time dependence. Predictions of a finite-parameter MIS-type model beyond the near-equilibrium regime are model-dependent, because higher-order terms are determined by how the model is completed. Attractor behavior can reduce sensitivity in some observables while leaving others informative. And in some classical field examples, suppressed sensitivity can later partially recover. In short, attractors are a useful organizing idea, but their scope and precise implications must be tested case by case.