On the Temperature Dependence of the Casimir Force for Bulk Lossy Media

TL;DR

This paper examines the limitations of the Lifshitz formula in describing how the Casimir force between lossy metals depends on temperature, emphasizing the importance of sample size and wavelength considerations.

Contribution

It identifies the conditions under which the Lifshitz theory fails due to finite sample sizes, clarifying the temperature dependence of the Casimir force in realistic scenarios.

Findings

Lifshitz's theory is invalid when fluctuation wavelengths exceed sample sizes.

Linearly decreasing temperature dependence occurs only in large or dirty samples at high temperatures.

The study resolves the inconsistency with the Nernst theorem by clarifying the limits of the linear dependence.

Abstract

We discuss the limitations of the applicability of the Lifshitz formula to describe the temperature dependence of the Casimir force between two bulk lossy metals. These limitations follow from the finite sizes of the interacting bodies. Namely, Lifshitz's theory is not applicable when the characteristic wavelengths of the fluctuating fields, responsible for the temperature-dependent terms in the Casimir force, is longer than the sizes of the samples. As a result of this, the widely discussed linearly decreasing temperature dependence of the Casimir force can be observed only for dirty and/or large metal samples at high enough temperatures. This solves the problem of the inconsistency between the Nernst theorem and the "linearly decreasing temperature dependence" of the Casimir free energy, because this linear dependence is not valid when $T \to 0$