Concerning verification of the Nernst theorem for Casimir-Lifshitz free energy

TL;DR

This paper examines the thermodynamic consistency of the Casimir-Lifshitz free energy by analyzing the real frequency axis, arguing that causal materials with temperature-independent permittivity do not violate Nernst's theorem, clarifying ongoing debates.

Contribution

It offers a new perspective by evaluating the Casimir free energy on the real frequency axis, highlighting conditions under which the Nernst theorem holds for causal materials.

Findings

Causal materials with temperature-independent permittivity do not violate Nernst's theorem.

The apparent violation arises from divergences when frequency and temperature vanish simultaneously.

The Nernst heat theorem is applicable only when theories are valid at zero temperature and frequency.

Abstract

By regarding the Lifshitz expression for the Casimir free energy on the real frequency axis rather than the imaginary Matsubara frequencies as is customary, new light is shed on the ongoing debate regarding the thermodynamical consistency of this theory in combination with common permittivity models. It is argued that when permittivity is temperature independent over a temperature interval including zero temperature, a cavity made of causal material with continuous dispersion properties separated by vacuum cannot violate Nernst's theorem (the third law of thermodynamics). The purported violation of this theorem pertains to divergencies in the double limit in which frequency and temperature vanish simultaneously. While any model should abide by the laws of thermodynamics within its range of applicability, we emphasise that the Nernst heat theorem is a relevant criterion for choosing amongst candidate theories only when these theories are fully applicable at zero temperature and frequency.