# On the high temperature limit of the Casimir energy

## TL;DR

This paper develops a method to derive explicit asymptotic expressions for the Casimir free energy at high temperatures, revealing classical and heat-kernel contributions, and explores implications for various geometries and thermodynamic laws.

## Contribution

The paper introduces a novel approach to compute high-temperature Casimir energies, including new numerical results and insights into thermodynamic violations in curved spaces.

## Key findings

- Casimir pressure on a rectangular tube can be twice that of parallel plates at high temperature.
- Nonzero mass terms violate the third law of thermodynamics for Casimir systems.
- Negative Casimir entropy observed on 3-torus and 3-sphere, indicating vacuum instability.

## Abstract

We introduce a useful approach to find asymptotically explicit expressions for the Casimir free energy at large temperature. The resulting expressions contain the classical terms as well as the few first terms of the corresponding heat-kernel expansion, as expected. This technique works well for many familiar configurations in Euclidean as well as non-Euclidean spaces. By utilizing this approach, we provide some new numerically considerable results for the Casimir pressure in some rectangular ideal-metal cavities. For instance, we show that at sufficiently large temperature, the Casimir pressure acting on the sidewalls of a rectangular tube can be up to twice that of the two parallel planes. We also apply this technique for calculating the Casimir free energy on a 3-torus as well as a 3-sphere. We show that a nonzero mass term for both scalar and spinor fields as well on the torus as on the sphere, violates the third law of thermodynamics. We obtain some negative values for the Casimir entropy on the 3-torus as well as on the 3-sphere. We speculate that these negative Casimir entropies can be interpreted thermodynamically as an instability of the vacuum state at finite temperatures