Moving mirrors and the fluctuation-dissipation theorem

TL;DR

This paper studies the motion of a mirror in a thermal scalar field, deriving the fluctuation-dissipation theorem and analyzing the mirror's diffusion behavior at finite and zero temperatures.

Contribution

It explicitly establishes the fluctuation-dissipation theorem for a moving mirror in (1+1) dimensions and analyzes the mirror's diffusion, including zero temperature logarithmic diffusion.

Findings

Standard diffusion at finite temperature

Logarithmic diffusion at zero temperature

Clarification of zero temperature limit behavior

Abstract

We investigate the random motion of a mirror in (1 + 1)-dimensions that is immersed in a thermal bath of massless scalar particles which are interacting with the mirror through a boundary condition. Imposing the Dirichlet or the Neumann boundary conditions on the moving mirror, we evaluate the mean radiation reaction force on the mirror and the correlation function describing the fluctuations in the force about the mean value. From the correlation function thus obtained, we explicitly establish the fluctuation-dissipation theorem governing the moving mirror. Using the fluctuation-dissipation theorem, we compute the mean-squared displacement of the mirror at finite and zero temperature. We clarify a few points concerning the various limiting behavior of the mean-squared displacement of the mirror. While we recover the standard result at finite temperature, we find that the mirror diffuses logarithmically at zero temperature, confirming similar conclusions that have been arrived at earlier in this context. We also comment on a subtlety concerning the comparison between zero temperature limit of the finite temperature result and the exact zero temperature result.