Self-similar equilibration of strongly interacting systems from holography

We study the equilibration of a class of far-from-equilibrium strongly interacting systems using gauge-gravity duality. The systems we analyze are $2+1$ dimensional and have a four-dimensional gravitational dual. A prototype example of a system we analyze is the equilibration of a two-dimensional fluid which is translational invariant in one direction and is attached to two different heat baths with different temperatures at infinity in the other direction. We realize such setup in gauge-gravity duality by joining two semi-infinite asymptotically anti-de Sitter (AdS) black branes of different temperatures, which subsequently evolve towards equilibrium by emitting gravitational radiation towards the boundary of AdS. At sufficiently late times the solution converges to a similarity solution, which is only sensitive to the left and right equilibrium states and not to the details of the initial conditions. This attractor solution not only incorporates the growing region of equilibrated plasma but also the outwardly propagating transition regions, and can be constructed by solving a single ordinary differential equation.

Figure 1

Cohomogeneity-1 similarity solutions to the Calabi-flow equation on the plane. The coordinate $\mu$, defined in (8), is a scale-invariant quantity, so the spatial profile is expanding with time. The solid curves show different values of $C$ which label the equilibrium state of the right-hand asymptotic system as a Dirichlet boundary condition (13). The values from left to right are $C=0.1$, 0.5, 1.0, 2.0, 4.0 and the dashed curve is the linear solution (12).

Figure 2

The similarity solution describing closely separated equilibria, after performing a certain Weyl transformation. Top panel: The function $j$ describing the deviation from equilibrium for the metric function $\sigma$. Middle panel: The energy density obtained from the eigenvalue equation, $\widehat{\delta ϵ}$, which shows the deviation from equilibrium and is defined in the text. The energy density is independent of time at fixed $x/t^{1/4}$ for each of the three flat space regions labeled by $(-)$, $(0)$, $(+)$. The central equilibrium region has a spatial extent which grows as $t^{1/4}$. Bottom panel: The time-rescaled difference between longitudinal and transverse pressures, $\widehat{\delta P}$, also defined in the text.

Figure 3

Evolution of the initial data (28) according to Eq. (4), showing convergence to the planar similarity solution in red, which is prescribed only by ${\sigma }_{\pm }$. Each curve shows a different time in the evolution.