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arXiv:2610.23354

Temperature dependence of the Casimir force between plane-layered structures: an approach based on the principle of argument

Michael V. Davidovich

quant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026
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Abstract

Based on the complex analysis theorem the principle of argument, a temperature-dependent formula for the Casimir-Lifshitz dispersion force of the interaction of two arbitrary plane-layered structures separated by either a vacuum gap of size d or a layer of size d consisting of an arbitrary dissipative dielectric with dispersion is derived. The formula consists of the temperature-dependent Lifshitz formula plus the temperature-dependent contour integral

Comments
30 pages, 3 figures, 34 references

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Based on the complex analysis theorem the principle of argument, a temperature-dependent formula for the Casimir-Lifshitz dispersion force of the interaction of two arbitrary plane-layered structures separated by either a vacuum gap of size d or a layer of size d consisting of an arbitrary dissipative dielectric with dispersion is derived. The formula consists of the temperature-dependent Lifshitz formula plus the temperature-dependent contour integral
Based on the van Kampen-Schram method, a formula for the Casimir force between two plane-layered structures is obtained, which includes the sum of the Matsubara frequencies and a contour integral. It is shown that the contour integral is not taken into account in the Lifshitz formula. Taking it into account gives a small contribution at high temperatures when the Lifshitz formula is approximately correct. However, at low temperatures, its contribution is significant, and the Lifshitz formula for the finite temperature does not transform into the Lifshitz formula at zero temperature. An example of interaction of metal plates is considered.1
The corrections to Casimir pressure (force) on two plates of real metal described by the dielectric constant of the Drude, Drude-Lorentz, Drude-Smith and plasma models at the finite temperature and conductivity are considered. We consider the contour integral lost in the Lifshitz formula and discuss inconsistencies for the corrections to the Casimir force given in the literature. Numerical and analytical results in the near and far zones are presented. The two first-order correction obtained due to the actual dielectric constant of the metal coincided with the literature data.
We present a time-domain scheme for computing Casimir forces within the Maxwell stress tensor formalism, together with a specific realization using the finite-element-based discontinuous Galerkin time-domain method. The approach enables accurate evaluation of Casimir--Lifshitz interactions for a wide range of geometries and material properties at finite temperature. At the core of the method, the electromagnetic Green's tensor is expressed as the system's response to dipolar excitations, thereby recasting the Maxwell stress tensor into a set of classical scattering problems driven by electric and magnetic dipoles. We validate the approach against reference calculations of the Casimir interaction between parallel half-spaces at both zero and nonzero temperature. We further demonstrate its applicability to finite, cylindrically symmetric geometries for which closed-form solutions are unavailable, obtaining accurate agreement with asymptotic predictions based on physical considerations. These findings illustrate the method's potential for studying Casimir interactions in realistic micro- and nanoscale structures, relevant to nanodevice design and experimental settings.
In this study, we consider motion of the massive and charged Bose gas trapped in electric and magnetic fields between two parallel plates in the x−yx-yx−y plane that are separated by a distance ddd in the zzz direction. We derive analytical expression of the grand canonical potential of the quantum particles by using of Ketterle and van Druten approximation. By evaluating canonical potential, we obtain Casimir potential and force at the Bose-Einstein condensation temperature TcT_{c}Tc​. We also show that Casimir force oscillates depends on distance ddd for fixed parameters.
Broadband optical response governs light-matter interactions across photonics, plasmonics, thermal radiation, and quantum fluctuation electrodynamics, yet determining a continuous dielectric function over many decades in frequency typically requires combining multiple spectroscopies, extrapolations, and material models. Here we show that quantum-fluctuation forces provide a route to broadband optical characterization. Casimir interactions depend on the dielectric response of materials across the electromagnetic spectrum, but this information is encoded through Lifshitz theory in a spectrally weighted and nontrivial way. By training physics-constrained supervised learning models on synthetic dielectric spectra and their corresponding Casimir force curves, we invert this relationship and constrain the complex permittivity of materials over more than seven orders of magnitude in frequency and provide an estimate of their values from force-distance data. The reconstruction reveals a direct separation-frequency correspondence: large separations constrain the free-carrier response, whereas shorter separations encode higher-frequency resonant structure. Applying the method to measured force gradients identifies the current experimental limits imposed by measurement noise, restricted separation range, and model complexity. These results establish fluctuation-induced forces as a spectrally weighted route to broadband optical characterization and define the experimental and physical limits that govern what spectral information is accessible from near-field quantum electromagnetic measurements.