Casimir Effect: The Classical Limit

TL;DR

This paper investigates the high-temperature limit of the Casimir effect, showing that the relative Casimir energy diminishes exponentially with temperature and that the classical limit of Casimir entropy depends solely on boundary geometry.

Contribution

It introduces the concept of relative Casimir energy and entropy, analyzing their behavior in the classical limit without imposing sharp boundary conditions, considering realistic boundary interactions.

Findings

Relative Casimir energy vanishes exponentially at high temperature.

Classical Casimir entropy approaches a finite value depending only on boundary geometry.

Casimir thermodynamical quantities exhibit logarithmic temperature deviations due to boundary self-energy.

Abstract

We analyze the high temperature (or classical) limit of the Casimir effect. A useful quantity which arises naturally in our discussion is the ``relative Casimir energy", which we define for a configuration of disjoint conducting boundaries of arbitrary shapes, as the difference of Casimir energies between the given configuration and a configuration with the same boundaries infinitely far apart. Using path integration techniques, we show that the relative Casimir energy vanishes exponentially fast in temperature. This is consistent with a simple physical argument based on Kirchhoff's law. As a result the ``relative Casimir entropy", which we define in an obviously analogous manner, tends, in the classical limit, to a finite asymptotic value which depends only on the geometry of the boundaries. Thus the Casimir force between disjoint pieces of the boundary, in the classical limit, is entropy driven and is governed by a dimensionless number characterizing the geometry of the cavity. Contributions to the Casimir thermodynamical quantities due to each individual connected component of the boundary exhibit logarithmic deviations in temperature from the behavior just described. These logarithmic deviations seem to arise due to our difficulty to separate the Casimir energy (and the other thermodynamical quantities) from the ``electromagnetic'' self-energy of each of the connected components of the boundary in a well defined manner. Our approach to the Casimir effect is not to impose sharp boundary conditions on the fluctuating field, but rather take into consideration its interaction with the plasma of ``charge carriers'' in the boundary, with the plasma frequency playing the role of a physical UV cutoff. This also allows us to analyze deviations from a perfect conductor behavior.