Brownian motion in AdS/CFT (0812.5112v3)

Abstract: We study Brownian motion and the associated Langevin equation in AdS/CFT. The Brownian particle is realized in the bulk spacetime as a probe fundamental string in an asymptotically AdS black hole background, stretching between the AdS boundary and the horizon. The modes on the string are excited by the thermal black hole environment and consequently the string endpoint at the boundary undergoes an erratic motion, which is identified with an external quark in the boundary CFT exhibiting Brownian motion. Semiclassically, the modes on the string are thermally excited due to Hawking radiation, which translates into the random force appearing in the boundary Langevin equation, while the friction in the Langevin equation corresponds to the excitation on the string being absorbed by the black hole. We give a bulk proof of the fluctuation-dissipation theorem relating the random force and friction. This work can be regarded as a step toward understanding the quantum microphysics underlying the fluid-gravity correspondence. We also initiate a study of the properties of the effective membrane or stretched horizon picture of black holes using our bulk description of Brownian motion.

Analysis of Brownian Motion in AdS/CFT

The paper "Brownian Motion in AdS/CFT," authored by Jan de Boer, Veronika Hubeny, Mukund Rangamani, and Masaki Shigemori, explores a novel perspective on Brownian motion through the lens of the AdS/CFT correspondence. This paper aims to advance the understanding of the quantum microphysics underlying the fluid-gravity correspondence and paves the way for insights into the dissipation processes in strongly coupled gauge theories.

The authors employ the AdS/CFT framework, specifically using a probe fundamental string in an asymptotically AdS black hole background, to model a boundary particle performing Brownian motion. In the bulk, the string's endpoint at the boundary corresponds to a heavy external particle within the boundary Conformal Field Theory (CFT). This setup allows them to paper the erratic motion induced by the black hole's environment, which thermally excites modes on the string.

Semiclassical Analysis

Semiclassically, the Hawking radiation emanating from the AdS black hole thermally excites the string, causing its endpoint in the boundary CFT to undergo Brownian motion. The paper verifies that this motion follows a generalized Langevin equation characterized by a memory kernel $\gamma(t)$ and an auto-correlation function $\kappa(t)$. Through this holographic model, the paper extracts the friction and random force coefficients relevant to the Langevin dynamics.

Critical numerical results are unveiled, such as the diffusion constant $D$ inversely proportional to the temperature $T$, which is consistent with previous results in strongly coupled plasmas. The authors identify key time scales associated with the motion: the relaxation time $t_{\text{relax}}$, the collision duration $t_{\text{coll}}$, and the mean free path time $t_{\text{mfp}}$. Notably, they find that $t_{\text{mfp}} \sim 1 / (\sqrt{\lambda} T)$, indicating many collisions occur simultaneously in a strongly-coupled environment, a result contrasting with the kinetic theory picture in weaker coupling regimes.

Theoretical Implications and Fluctuation-Dissipation Theorem

An intriguing aspect of the paper is the bulk examination of the fluctuation-dissipation theorem, traditionally derived from linear response theory. The authors provide a bulk-side proof, demonstrating the consistency of the boundary Langevin dynamics with the fluctuation-dissipation relations. This insight emphasizes that bulk black hole dynamics can encapsulate nuanced equilibrium features of thermal field theories.

Future Directions

While the authors primarily focus on non-relativistic Langevin dynamics, extending this framework to incorporate relativistic effects may offer deeper insights, particularly as the dynamics of relativistic Brownian motion remains less understood. Furthermore, exploring connections with rapidly scrambling systems—a concept gaining attention due to its implications for black hole information processing—could illuminate new aspects of holographic field theories.

The granularity of the stretched horizon, conceptualized through the membrane paradigm, offers another intriguing avenue. The paper's derivations suggest that the structure of quasi-particles on the horizon may parallel the dynamics observed in boundary diffusion processes, prompting further investigation into the microscopic nature of these quasi-particles.

This paper represents an essential step in reconciling holographic models with conventional physics of dissipative systems, offering promising directions for theoretical advancements in string theory and condensed matter physics.