Thermal Noise and Stochastic Strings in AdS/CFT

TL;DR

The paper provides a concrete, holographically grounded construction of thermal noise for a heavy quark in N=4 SYM via AdS/CFT. By employing the Kruskal extension and the Keldysh contour, it derives a stochastic boundary condition on the stretched horizon and shows how horizon fluctuations propagate to the boundary, producing a generalized Langevin equation with a fluctuation-dissipation consistent noise. It confirms this structure through bulk-to-bulk correlators and KMS relations, and offers a physical picture in terms of a flip-flopping trailing string, with implications for gravitons and other bulk fields. The results solidify the connection between dissipation and fluctuations in holographic plasmas and provide a framework for out-of-equilibrium fluctuation analyses.

Abstract

We clarify the structure of thermal noise in AdS/CFT by studying the dynamics of an equilibrated heavy quark string. Using the Kruskal extension of the correspondence to generate the dynamics of the field theory on the Keldysh contour, we show that the motion of the string is described by the classical equations of motion with a stochastic boundary condition on the stretched horizon. The form of the stochastic boundary condition is consistent with the dissipation on this surface and is found by integrating out the fluctuations inside of the stretched horizon. Solving the equations of motion for the fluctuating string we determine the full frequency dependence of the random force on the boundary quark and show that it is consistent with the frequency dependent dissipation. We show further that the stochastic motion reproduces the bulk to bulk two point functions of the Kruskal formalism. These turn out to be related to the usual retarded bulk to bulk propagator by KMS relations. Finally we analyze the stochastic equations and give a bulk picture of the random boundary force as a flip-flopping trailing string solution. The basic formalism can be applied to the fluctuations of gravitons, dilatons, and other fields.

Figures (6)

• Figure 1: 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 $r_m$ with $r_m \gg 1$. Gravity pulls downward in this figure.
• Figure 2: The Schwinger Keldysh contour. The fields labeled by "1" live on the upper time ordered axis, while the fields labeled by "2" live on the lower anti-time ordered axis.
• Figure 3: The full Kruskal plane. The right quadrant corresponds to the amplitude of the field theory (the "1" axis) while the left quadrant corresponds to the conjugate amplitude of the field theory (the "2" axis).
• Figure 4: Balance of forces on the stretched horizon. The resistive force $-\eta \dot{x}^h$ precisely balances the random force $\xi^h$ and the tension ${T_o}$ leading to overdamped motion.
• Figure 5: (a) The physical picture that emerges when observing the quark on relatively short time scales $1/T \ll \tau_{\rm obs} \ll \tau_R$. Here we show three subsequent time steps, $t_1, t_2,t_3$; at each time step the string fluctuates to a new "trailing string" giving rise to a random force on the boundary. The average trailing string is perceived as a drag. (b) The physical picture that emerges on very long time and spatial scales. The horizon diffuses and the string is brought along.