Out of equilibrium thermal Casimir effect in a model polarizable material

TL;DR

This paper studies the out of equilibrium relaxation dynamics of the thermal Casimir force in dielectric materials, revealing how the force approaches equilibrium with different decay behaviors depending on the relaxation time distribution.

Contribution

It introduces a model for dielectric relaxation affecting the thermal Casimir force, showing how the force's amplitude evolves over time and highlighting differences in decay rates for various dielectric relaxation spectra.

Findings

Force amplitude increases over time to equilibrium

Exponential relaxation in single relaxation time systems

Power law decay in broad relaxation time distributions

Abstract

Relaxation of the thermal Casimir or van der Waals force for a model dielectric medium is investigated. We start with a model of interacting polarization fields with a dynamics that leads to a frequency dependent dielectric constant of the Debye form. In the static limit the usual zero frequency Matsubara mode component of the Casimir force is recovered. We then consider the out of equilibrium relaxation of the van der Waals force to its equilibrium value when two initially uncorrelated dielectric bodies are brought into sudden proximity. It is found that the spatial dependence of the out of equilibrium force is the same as the equilibrium one but it has a time dependent amplitude, or Hamaker coefficient, which increases in time to its equilibrium value. The final relaxation to the equilibrium value is exponential in systems with a single or finite number of polarization field relaxation times. However, in systems, such as those described by the Havriliak-Negami dielectric constant, with a broad distribution of relaxation times, we observe a much slower power law decay to the equilibrium value.