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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

Submission summary

As Contributors: Gert-Ludwig Ingold · Tanja Schoger
Arxiv Link: (pdf)
Date accepted: 2021-04-19
Date submitted: 2021-02-08 20:52
Submitted by: Ingold, Gert-Ludwig
Submitted to: SciPost Physics
Academic field: Physics
  • Combinatorics
  • Atomic, Molecular and Optical Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


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.

Published as SciPost Phys. Core 4, 011 (2021)

Author comments upon resubmission

List of changes

List of Changes

1) We have rewritten the first paragraph of the introduction according to
requests 2 and 3 of Referee 2. A detailed description of our changes can
be found in our reply to the second report.

2) The former third paragraph of the introduction is now split into two
paragraphs (three and four in version 2). As a response to the weakness
mentioned by the second referee, we extended the discussion of the importance
of our result for experiments in paragraph 3. In paragraph 4, we rephrased
the role of the zero-frequency term in semi-analytical approaches. For more
details, we refer to our reply on the fourth request of referee 2.

3) We have added a paragraph (paragraph 5 in version 2) as a response to
request 1 of referee 1, where we discuss in more detail how by combining
various results from the literature, our result could be obtained.

4) In paragraph 6, the former fourth paragraph, we extended the motivation
for our calculation.

5) In the last paragraph of the introduction, we specified what we mean by
'dual result'. More details can be found in our response to request 5 by
referee 2. Furthermore, we added the adjective 'spherical' to the monopole
contributions, to avoid confusion with the bispherical multipoles which,
in the new version, are now mentioned beforehand.

6) In the first paragraph of section 2, we introduced the definition of a
Drude metal by specifying the corresponding dielectric function (see response
to request 7 of referee 2).

7) In paragraph two of section 2, we added a remark that the scattering approach
to the Casimir effect results in a sum, where a round-trip operator is
evaluated at the Matsubara frequencies.

8) Before the old eq. (4), now (5) we removed 'the Wick rotated frequency $\xi$',
since we already introduced imaginary frequencies as a consequence of point 6.

9) Before the old eq. (8), now (9) we removed 'Making use of the symmetries of
cosine and hyperbolic cosine, we obtain' as a response to request 8 of referee 2.

10) At the beginning of the last paragraph in section 2, we used the singular form
for the monopole terms, more precisely, we replaced 'monopole terms do' by
'the monopole term does' and 'monopole terms' by 'monopole term'. This change
was made to be consistent with the discussion after eq. (7), now (8).
Moreover, we consistently replaced 'monopole contribution' with the plural form
'monopole contributions'.

11) At the end of the last paragraph in section 2, we replaced 'found to be dual'
by 'equivalent' (cf. request 9 of referee 2).

12) In the last paragraph of section 3, we specified the possible numerical
advantage of our result by replacing 'is' with 'may be numerically'. Moreover,
as a response to request 11 of referee 2, we replaced 'the sum is converging
considerably faster' by 'it possesses better convergence properties'.

13) We extended the first paragraph of section 4.1, as discussed in our reply to
request 13 of referee 2.

14) Before the old eq. (24), now (25) we replaced 'monopole contribution' with
'monopole contributions' (cf. point 10).

15) In the caption of fig. 2, we included a reference to equation (26), former
eq. (25). Moreover, we specified the shown block matrix by defining the
corresponding values of r and k.

16) Before eq. (27), now (28) and after eq. (32), now (33), we replaced
'monopole contribution' with 'monopole contributions' (cf. point 10).

17) In response to request 2 of referee 1, we extended the discussion in the
first part of section 4.4. More precisely, we added the result by Fosco
et al., which is now given in eq. (46). Furthermore, we adopted the choice
of units by Fosco, a fact now also stated in a footnote, and changed the
definitions of the capacitance coefficients (45)-(47), now (47)-(49).
Below these definitions, we highlighted the relation between the capacitance
coefficients and the round-trip description. We also added the historic
reference [27].

18) For consistency with the changes mentioned in point 17, we replaced '$T^2$'
by '$R_1R_2T^2$, in the last paragraph of section 4.4.

19) Before eq. (53), now (55), we corrected a typo by replacing 'reads' with 'read'.

20) Before eq. (58), now (60), after eq. (64), now (66) and after eq. (74), now (76),
we replaced 'monopole contribution' with 'monopole contributions' (cf. point 10).

21) After eq. (88), now (90) and after eq. (100), now (102) we replaced
'monopole contribution' with 'monopole contributions' (cf. point 10).

Reports on this Submission

Anonymous Report 2 on 2021-4-1 Invited Report

  • Cite as: Anonymous, Report on arXiv:2009.14090v2, delivered 2021-04-01, doi: 10.21468/SciPost.Report.2673


I maintain the previous ones:
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 calculations are rather complex and difficult to follow.


The relation with previous works has been clarified. In my opinion, the paper meets the acceptance criteria of the journal. I recommend publication in its present form.

Requested changes

No further suggestions.

  • validity: top
  • significance: high
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2021-2-26 Invited Report


The authors have revised the paper and modified the manuscript in a completely satisfactory manner as far as I am concerned and the manuscript is I believe suitable for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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