Casimir effect for a dilute dielectric ball at finite temperature

TL;DR

This paper calculates the finite-temperature Casimir effect for a dilute dielectric sphere, deriving thermodynamic properties and analyzing their behavior at temperature extremes using advanced mathematical techniques.

Contribution

It provides a rigorous derivation of the internal and free energies for a dielectric sphere at finite temperature, extending zero-temperature Casimir calculations with a new renormalization approach.

Findings

Thermodynamic quantities are explicitly calculated at low and high temperatures.

A closed-form expression for the sum over angular momentum is obtained.

Divergences are successfully removed through a specialized renormalization procedure.

Abstract

The Casimir effect at finite temperature is investigated for a dilute dielectric ball; i.e., the relevant internal and free energies are calculated. The starting point in this study is a rigorous general expression for the internal energy of a system of noninteracting oscillators in terms of the sum over the Matsubara frequencies. Summation over the angular momentum values is accomplished in a closed form by making use of the addition theorem for the relevant Bessel functions. For removing the divergences the renormalization procedure is applied that has been developed in the calculation of the corresponding Casimir energy at zero temperature. The behavior of the thermodynamic characteristics in the low and high temperature limits is investigated.