Casimir force between two ideal-conductor walls revisited

Abstract

PACS. 05.20.Jj -Statistical mechanics of classical fluids. PACS. 12.20.-m -Quantum electrodynamics. PACS. 11.10.Wx -Finite-temperature field theory.Abstract. -The high-temperature aspects of the Casimir force between two neutral conducting walls are studied. The mathematical model of "inert" ideal-conductor walls, considered in the original formulations of the Casimir effect, is based on the universal properties of the electromagnetic radiation in the vacuum between the conductors, with zero boundary conditions for the tangential components of the electric field on the walls. This formulation seems to be in agreement with experiments on metallic conductors at room temperature. At high temperatures or large distances, at least, fluctuations of the electric field are present in the bulk and at the surface of a particle system forming the walls, even in the high-density limit: "living" ideal conductors. This makes the enforcement of the inert boundary conditions inadequate. Within a hierarchy of length scales, the high-temperature Casimir force is shown to be entirely determined by the thermal fluctuations in the conducting walls, modelled microscopically by classical Coulomb fluids in the Debye-Hückel regime. The semi-classical regime, in the framework of quantum electrodynamics is studied in the companion letter by P.R.Buenzli and Ph.A.Martin [1].This letter is related to the one by Buenzli and Martin [1]. For the sake of completeness, we cannot avoid repeating a few things.Casimir showed in his famous paper [2] that fluctuations of the electromagnetic field in vacuum can be detected and quantitatively estimated via the measurement of a macroscopic attractive force between two parallel neutral metallic plates; for a nice introduction to the Casimir effect see [3] and for an exhaustive review see [4].Let us recall briefly, within the formalism of Ref.[3], some aspects of the usual theory for plates considered as made of ideal conductors, which are relevant in view of the present letter. We consider the 3D Cartesian space of points r = (x, y, z) where a vacuum is localized in the subspace Λ = {r|x ∈ (−d/2, d/2); (y, z) ∈ R 2 } between two ideal-conductor walls (thick slabs) at a distance d from each other. The time-dependent electric E(r, t) and magnetic B(r, t) fields in Λ are the solutions of the Maxwell equations in vacuum, subject to the boundary

Analytical and numerical verification of the Nernst theorem for metals

“…And there are other papers in agreement with ours: for instance, Jancovici andŠamaj [14], and Buenzli and Martin [15] considered the Casimir force between two plates in the high-temperature limit. They found that the linear dependence in T for the Casimir force to be reduced by a factor of two from the behavior of of an ideal metal, this being a signal of the vanishing influence from the zero-frequency TE mode.…”

Casimir energies: Temperature dependence, dispersion, and anomalies

“…In particular, the lack of a thermodynamic inconsistency has been conclusively demonstrated [33,34], by showing that the free energy for a Casimir system made from real metal plates with impurities has a quadratic temperature dependence at low temperature. Further evidence for the validity of the notion of excluding the TE zero mode for metals comes from the recent work of Buenzli and Martin [35], corroborating earlier work by these authors and others [36,37], who show from a microscopic viewpoint that the high-temperature behavior of the Casimir force is half that of an ideal metal, a rather conclusive demonstration that the TE zero mode is not present. 1 Our purpose with the present paper is not to study the temperature corrections in further detail.…”

Present status of controversies regarding the thermal Casimir force

“…At large separations or high temperatures the Drude model approach predicts 1/2 the magnitude of the Casimir free energy and force than between ideal metals. (The same prediction was obtained using the model of a conducting wall by the classical Coulomb fluid [16] or 'nonrelativistic quantum electrodynamics' [17].) This prediction is in contradiction with the classical limit based on Kirchhoff's law [18].…”