Finite Temperature Casimir Effect for a Dilute Ball Satisfying $\epsilon \mu=1$

TL;DR

This paper calculates the finite temperature Casimir free energy for a dielectric sphere with the condition , using Green function and mode summation methods, and explores effects of different dispersion relations.

Contribution

It provides the first detailed analysis of the finite temperature Casimir effect for a dielectric ball satisfying , including both simple and spatial dispersion models.

Findings

High temperature Casimir free energy is negative and proportional to .

Green function and mode summation methods yield equivalent results.

Dispersion relations significantly influence the Casimir free energy.

Abstract

The finite temperature Casimir free energy is calculated for a dielectric ball of radius $a$ embedded in an infinite medium. The condition $\epsilon\mu=1$ is assumed for the inside/outside regions. Both the Green function method and the mode summation method are considered, and found to be equivalent. For a dilute medium we find, assuming a simple "square" dispersion relation with an abrupt cutoff at imaginary frequency $\hat \omega= \omega_0$, the high temperature Casimir free energy to be negative and proportional to $x_0 \equiv \omega_0 a$. Also, a physically more realistic dispersion relation involving spatial dispersion is considered, and is shown to lead to comparable results.