Fluctuation, dissipation, and thermalization in non-equilibrium AdS_5 black hole geometries

TL;DR

<3-5 sentence high-level summary>We address how dissipation and fluctuations arise in non-equilibrium AdS$_5$ black hole geometries by formulating Hawking radiation as an initial-value problem and encoding it in a horizon effective action. The authors introduce horizon correlators on the stretched horizon, show how they propagate into the bulk to determine bulk and boundary correlators, and derive a bulk fluctuation-dissipation theorem from horizon data. In equilibrium this reproduces heavy-quark Brownian motion, and out of equilibrium the Wigner transforms of correlators obey FDT at high frequency, providing a practical route for numerical computation of thermalization dynamics in holographic plasmas. The framework unifies near-horizon quantum fluctuations with boundary observables, offering a quantum generalization of the membrane paradigm and a tool for studying non-equilibrium thermalization in strongly coupled gauge theories.

Abstract

We give a simple recipe for computing dissipation and fluctuations (commutator and anti-commutator correlation functions) for non-equilibrium black hole geometries. The recipe formulates Hawking radiation as an initial value problem, and is suitable for numerical work. We show how to package the fluctuation and dissipation near the event horizon into correlators on the stretched horizon. These horizon correlators determine the bulk and boundary field theory correlation functions. In addition, the horizon correlators are the components of a horizon effective action which provides a quantum generalization of the membrane paradigm. In equilibrium, the analysis reproduces previous results on the Brownian motion of a heavy quark. Out of equilibrium, Wigner transforms of commutator and anti-commutator correlation functions obey a fluctuation-dissipation relation at high frequency.

Figures (6)

• Figure 1: Two figures from Ref. Son:2009vu which motivate this work. (a) A schematic of a classical string in AdS$_{5}$ corresponding to a heavy quark. The horizon is at $r=1$ in the coordinates of this work. The stretched horizon is at $r_h=1+\epsilon$ and the endpoint of the string is at the boundary $r_m$ with $r_m \gg 1$. Gravity pulls downward in this figure. (b) Hawking radiation from the horizon induces stochastic motion of the string in the bulk which we show for three subsequent time steps, $t_1, t_2,t_3$. The random string configurations give rise to a random force in the boundary theory. The Hawking radiation is encoded in an effective action on the stretched horizon $r_{h} = 1+ \epsilon$. The string fluctuations are small, $x_{\rm obs} \sim 1/\lambda^{1/4} T$.
• Figure 2: A congruence of outgoing null radial geodesics starting at time $t_0$. Qualitative insight on the propagation of initial data specified on the slice at $t = t_0$ can be understood from the congruence. Generic geodesics reach the boundary in a time $\Delta t \sim 1/T$. Initial data propagated on such trajectories reflects from the boundary and falls into the horizon with an infall time of order $1/T$. Geodesics originating exponentially close to the horizon take much longer to escape. Consequently, at late times the above-horizon geometry is filled with geodesics emanating exponentially close to the horizon at $t = t_0$. Because of this, the only initial data relevant at late times consists of the initial data exponentially close to the horizon at $t = t_0$.
• Figure 3: The composition law for retarded Green functions. The stretched horizon separates the two steps of the evolution. From the perspective of an observer in the exterior, the strip between the horizon at $r = 1$ and the stretched horizon at $r = 1 + \epsilon$ produces a simple horizon effective correlator, which acts like a source of radiation from the stretched horizon.
• Figure 4: Feynman graph used for computing the symmetrized correlation function $G_{rr}(\omega, r_1, r_2)$, see Eq. (\ref{['gsym']}).
• Figure 5: The Keldysh contour from an initial time $t_{o}$ where the density matrix is specified to a final time, $t_{\rm max}$. $x_{a}(t,r) = x_1(t,r) - x_2(t,r)$ must be zero at $t_{\rm max}$.