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Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach
by Tanja Schoger, Gert-Ludwig Ingold
This is not the current version.
|As Contributors:||Gert-Ludwig Ingold · Tanja Schoger|
|Arxiv Link:||https://arxiv.org/abs/2009.14090v1 (pdf)|
|Date submitted:||2020-09-30 08:03|
|Submitted by:||Ingold, Gert-Ludwig|
|Submitted to:||SciPost Physics|
We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.
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Anonymous Report 2 on 2020-12-22 Invited Report
1 The result appears to be a significant generalisation of for the thermal Casimir interaction between two Drude spheres of differing radii.
2 The calculation is very involved but the final result is quite elegant.
3 The paper recovers previously known results as special cases
1 The paper is aimed at a very narrow audience, not even at the general Casimir community
2 The presentation should be improved
The authors compute the thermal component of the Casimir free energy between two Drude spheres of unequal radii using the scattering approach. They carry out their calculation in the plane wave basis and find a formula which agrees with the results for the sphere plane set up and for two spheres of equal radii. The paper is very technical and some aspects should be made clearer to make the paper accessible to a more general readership. The authors use a method based on correcting the problem of a scaler field with Dirichlet boundary conditions to obtain the solution for the Drude problem. The difference between the elements in the expansion for the free energy are seen clearly between equations (8) and (9). However the computation is not straightforward and requires considerable combinatorial analysis. I think that the results of the paper are interesting. Although the calculation is very long to check it agrees with known results in certain established limits and the authors claim to have tested it numerically (although this is not shown explicitly). The paper thus seems suitable for publication.
1 The general level of English could be improved and the introductory part of the paper needs some improvement as it stands it needs a bit more explanation. See the points below.
2 P1 However, also thermal photons contribute to the Casimir force which survives the classical limit
This could be made clearer eg Thermal photons also contribute to the Casimir force. The authors could perhaps say here that the zero frequency Matsubara term is non zero and yields the thermal component of the Casimir force which is nonzero.
3 P1 Then, the free energy does no longer depend on Planck’s constant and is found to be linear in temperature - should be rewritten given the comments made about the previous sentence
4 P1 Here, the terms for non-zero Matsubara frequencies are treated within the derivative expansion while this approximation is less accurate for zero frequency. An exact analytical high-temperature expression will thus be valuable.
These two phrases are confused/not clear
5 P1 … of a scalar field which is found to be dual to the known result .. what do the authors mean by dual. The sentence is too concise to convey any meaning so it should either be expanded on or discussed later. Later on we see that the authors rederive the result of  but their result is given in a different form. I would thus say their result is equivalent to that of  in this case.
6 P3 Within the scattering approach to the Casimir effect , the free energy can be expressed as a Matsubara sum - this is a bit misleading, in any approach the free energy is expressed this way, and it is more common to say a sum over Matsubara frequencies.
7 P3 It would help the reader to give the definition of a Drude type metal where it is mentioned - I think this should be discussed in a detailed way as it is a crucial point in the paper.
8 P5 Making use of the symmetries of cosine and hyperbolic cosine - I am not sure what is meant by this
9 P5 again the same mysterious sentence - Casimir free energy which is found to be dual to the known result .
10 P5 After having determined the matrices associated with the bilinear forms in the exponentials, our main task will be to evaluate the corresponding determinants - I guess he authors mean the integrals in equation 8 which can be written as a determinant ?
11 P7 because the sum is converging considerably faster - should be the sum converges faster (I am not sure considerably should be used without more justification).
12 P7 after eq 19 it would be helpful to explain the difference in boundary conditions for the non expert reader if it has not already be done in response to the point above.
13 P7 To obtain the result for the electromagnetic case from the scalar case, we need to determine the contribution of all terms which contain at least one factor −1 when the product in (8) is expanded
- I see what this means but it should expressed more precisely.
Anonymous Report 1 on 2020-11-30 Invited Report
1. The paper contains a detailed analytical calculation of the Casimir free-energy for the geometry of two Drude spheres of arbitrary radii, in the high-temperature limit.
2. An interesting connection between Casimir physics and electrostatics is highlighted.
1. The main result can be obtained from existent results in the literature, so the complex calculations presented in the paper could be avoided. There is no clear justification for using an alternative approach based on the scattering formalism with a plane-wave basis.
In this paper, the authors compute the Casimir free energy for two Drude spheres of arbitrary radii, in the high-temperature limit, using the scattering approach with a plane-wave basis. The main results are given by Eq. (15), which gives the free energy for Dirichlet spheres, and Eq.(40), which gives the difference between the free energy for Drude and Dirichlet boundary conditions. The combination of both results gives an analytical expression for the Casimir free energy for Drude spheres, which is shown to reproduce correctly the known cases of a sphere in front of a plane, and two spheres of equal radii. Moreover, it reproduces the proximity force approximation in the short-distance expansion. The paper is clearly written and describes a technically complex calculation.
My main concern is the following: Eq. (15) has been previously obtained in Ref., while Eq.(40) is a particular case of a general result obtained in Ref.. If the main aim of the authors is to obtain an analytical result for the sphere-sphere Casimir free energy, many calculations could be avoided. If, instead, their main goal is to reproduce previous results using an alternative method, they should state this from the very beginning, and provide motivations to do that.
1. In the Introduction, the authors should mention that although the analytical formula for Drude spheres of different radii could be obtained from the results of Refs.  and , they will obtain the formula using an alternative approach, providing the motivations for doing that.
2. In Section 4.4, the relation between Eq.(40) and the general result of Ref. should be clearly stated. The quantity $\Delta$ is formally evaluated in  for a system on $N$ conductors of arbitrary shapes, and involves the logarithm of the determinant of the capacitance matrix. When evaluated for two spheres, this formula, which could be written in the paper, reproduces Eq.(40), up to terms not relevant for the evaluation of the force. I think this would reinforce the interesting connection between Casimir physics and electrostatics mentioned by the authors in the Conclusions.