# Recent Results on Thermal Casimir Force between Dielectrics and Related Problems

## TL;DR

This paper reviews recent advances in understanding the thermal Casimir force between various materials, emphasizing thermodynamic consistency, the impact of dielectric properties, and the relevance of Lifshitz theory to experimental results.

## Contribution

It clarifies the conditions under which Lifshitz theory aligns with thermodynamics and discusses the effects of dielectric properties and conductivity on the Casimir force.

## Key findings

- Lifshitz theory is thermodynamically consistent for dielectrics with finite static permittivity.
- Including dc conductivity in dielectrics violates the Nernst heat theorem.
- Experimental results support the use of frequency dispersion without spatial dispersion considerations.

## Abstract

We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the low-temperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat theorem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivity of dielectric plates is proved to lead to a violation of the Nernst heat theorem and, thus, is not related to the phenomenon of dispersion forces. The low-temperature asymptotics of the Casimir free energy, pressure and entropy are derived also in the configuration of one metal and one dielectric plate. The results are shown to be consistent with thermodynamics if the dielectric plate possesses a finite static dielectric permittivity. If the dc conductivity of a dielectric plate is taken into account this results in the violation of the Nernst heat theorem. We discuss both the experimental and theoretical results related to the Casimir interaction between metal and semiconductor with different charge carrier density. Discussions in the literature on the possible influence of spatial dispersion on the thermal Casimir force are analyzed. In conclusion, the conventional Lifshitz theory taking into account only the frequency dispersion remains the reliable foundation for the interpretation of all present experiments.