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

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.Comment: 28 pages, 6 figure

Time dependent diffusion in a disordered medium with partially absorbing walls: A perturbative approach

We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick (J. Chem. Phys. 76, 4255) we obtain a perturbative expansion for the time dependent particle density using volume fraction $f$ of spheres as an expansion parameter. The exact single particle $t$-operator for partially absorbing boundary condition is used to obtain a closed form time-dependent diffusion coefficient $D(t)$ accurate to first order in the volume fraction $f$. Short and long time limits of $D(t)$ are checked against the known short-time results for partially or fully absorbing boundary conditions and long-time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be $D(t)=5 a^2/(12 f D_{0} t) +O((D_0t/a^2)^{-2})$, to the first order of perturbation theory. Here $f$ is small but non-zero, $D_0$ the diffusion coefficient in the absence of spheres, and $a$ the radius of the spheres. The validity of this perturbative result is discussed