Casimir effect at nonzero temperature for wedges and cylinders

TL;DR

This paper analyzes the Casimir-Helmholtz free energy at nonzero temperature for cylindrical and wedge geometries with perfect conductors, deriving regularized energy expressions and confirming known zero and high temperature limits.

Contribution

It provides a regularized finite energy expression at nonzero temperature for wedges and cylinders, including corner singularities, and clarifies the temperature dependence through a non-dimensional parameter.

Findings

Zero temperature results are recovered as temperature approaches zero.

High temperature asymptotics for cylindrical shells are exactly reproduced.

The energy regularization accounts for corner singularities in wedges.

Abstract

We consider the Casimir-Helmholtz free energy at nonzero temperature $T$ for a circular cylinder and perfectly conducting wedge closed by a cylindrical arc, either perfectly conducting or isorefractive. The energy expression at nonzero temperature may be regularized to obtain a finite value, except for a singular corner term in the case of the wedge which is present also at zero temperature. Assuming the medium in the interior of the cylinder or wedge be nondispersive with refractive index $n$, the temperature dependence enters only through the non-dimensional parameter $2\pi naT$, $a$ being the radius of the cylinder or cylindrical arc. We show explicitly that the known zero temperature result is regained in the limit $aT\to 0$ and that previously derived high temperature asymptotics for the cylindrical shell are reproduced exactly.