Nonequilibrium effects in the Casimir force between two similar metallic plates kept at different temperatures

G.-L. Ingold, G. L. Klimchitskaya, and V. M. Mostepanenko

Phys. Rev. A 101, 032506 – Published 11 March 2020

Abstract

We study the Casimir pressure between two similar plates of finite thickness kept at different temperatures in the case when the dielectric permittivity of the plates depends on temperature. It is suggested to consider the dielectric permittivity at two different temperatures as the permittivities of two dissimilar bodies, thus, allowing to apply the theory of Casimir forces out of thermal equilibrium developed earlier in the literature. Following this approach, we show that, in addition to the equilibrium contribution to the nonequilibrium Casimir pressure, a proper nonequilibrium contribution arises for temperature-dependent dielectric permittivities. Furthermore, the equilibrium contribution in this case does not equal the mean of the equilibrium Casimir pressures at the temperatures of the plates. As an application, the total nonequilibrium Casimir pressure between two gold plates and between two titanium plates is calculated as a function of the plate thickness and their separation using the Drude and the plasma models. For plate separations ranging from 0.5 to $2 μ m$ , the relative difference between the theoretical predictions for these two models reaches 39%. The proper nonequilibrium term may be as large as 4% of the magnitude of the total nonequilibrium pressure.

Received 17 December 2019
Accepted 18 February 2020

DOI:https://doi.org/10.1103/PhysRevA.101.032506

Physics Subject Headings (PhySH)

Casimir effect & related phenomena Van der Waals interaction

Optical microcavities Quantum cavities

Polarizability

First-principles calculations Quantum correlations, foundations & formalism

Atomic, Molecular & Optical

Authors & Affiliations

G.-L. Ingold ¹, G. L. Klimchitskaya ^2,3, and V. M. Mostepanenko ^2,3,4

¹Universität Augsburg, Institut für Physik, 86135 Augsburg, Germany
²Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, Saint Petersburg 196140, Russia
³Institute of Physics, Nanotechnology and Telecommunications, Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg 195251, Russia
⁴Kazan Federal University, Kazan 420008, Russia

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Issue

Vol. 101, Iss. 3 — March 2020

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Images

Figure 1
The configuration of two parallel plates of thickness $d$ which are kept at different temperatures $T_{1}$ (as in the environment) and $T_{2}$ .
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Figure 2
The magnitudes of the nonequilibrium Casimir pressure are shown as functions of separation (a) for Au plates by the top and bottom lines computed using the plasma and Drude models, respectively, and (b) for Ti plates by the pairs of lines labeled 1 and 2 computed for the plates of $d = 20 − nm$ and $1 − μ m$ thicknesses, respectively, where, in each pair, the upper and lower lines are obtained using the plasma and Drude models, respectively.
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Figure 3
The nonequilibrium Casimir pressures normalized to the Casimir pressure between two ideal metal plates at zero temperature are shown as functions of separation (a) for Au plates by the top and bottom pairs of solid lines computed using the plasma and Drude models, respectively, where, in each pair, the lower and upper solid lines are for plate thicknesses of $d = 20 nm$ and $1 μ m$ , respectively, and (b) for Ti plates by the pairs of solid lines labeled 1 and 2 computed for the plate thicknesses of $d = 20 nm$ and $1 μ m$ , respectively, where, in each pair, the upper and lower solid lines are obtained using the plasma and Drude models, respectively. In all cases, the dashed lines are computed by means of the Drude model with the temperature-independent relaxation parameter taken at 300 K.
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Figure 4
The relative difference between the modified and the standard equilibrium contributions to the nonequilibrium Casimir pressure between two similar metallic plates described by the Drude model with the temperature-dependent relaxation parameter as a function of separation is shown in percentages by the four lines from top to bottom for Au plates with $d = 1 − μ m$ and 20-nm thicknesses and for Ti plates with $d = 1 μ m$ and 20 nm, respectively.
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Figure 5
The ratios of the proper nonequilibrium contribution and the total Casimir nonequilibrium pressure are shown in percentages as a function of the separation by the two solid lines from top to bottom for (a) Au plates with $d = 1 − μ m$ and 20-nm thicknesses, respectively, and (b) Ti plates with $d = 20 nm$ and $1 μ m$ , respectively.
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Figure 6
The proper nonequilibrium contribution to the Casimir pressure is shown as a function of plate thickness by the top and bottom pairs of lines computed at separations $a = 0.5$ and $1 μ m$ between the plates, respectively. In each pair, the upper line corresponds to Au plates whereas the lower line is for Ti plates.
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