Curvature driven diffusion, Rayleigh-Plateau, and Gregory-Laflamme

TL;DR

This paper explores the nonlinear dynamics of the Rayleigh-Plateau instability across dimensions, revealing phase transitions and potential analogies with black string instabilities in gravity.

Contribution

It demonstrates the connection between membrane and black hole instabilities, analyzing the phase structure and final states near the critical dimension using surface diffusion equations.

Findings

Unstable cylinders either pinch off or stabilize depending on initial conditions.

Near the critical dimension, the final state can be a non-uniform black string.

Surface diffusion models describe the dynamics and phase transitions.

Abstract

It can be expected that the respective endpoints of the Gregory-Laflamme black brane instability and the Rayleigh-Plateau membrane instability are related because the bifurcation diagrams of the black hole-black string system and the liquid drop-liquid bridge system display many similarities. In this paper, we investigate the non-linear dynamics of the Rayleigh-Plateau instability in a range of dimensions, including the critical dimension at which the phase structure changes. We show that near the critical dimension and above, depending on a parameter in initial conditions an unstable cylinder will either pinch off or converge to an equilibrium state. The equilibrium state is apparently non-uniform but has a constant mean curvature everywhere. The results suggest that in the gravity side, near the critical dimension and above, the final state of an unstable black string (which is not too long) is a non-uniform black string. The equation of motion adopted to describe the dynamics is the surface diffusion equation, which was originally proposed to describe a grooving process of heated metal surfaces. An interesting correspondence between the diffusion dynamics and black hole (thermo)dynamics is discussed.