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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

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For submitted/published material: please contact to point out this error

Dear editor,

we would like to thank you and the referees for assessing our manuscript. We
have prepared a revised version of the manuscript which accounts for the points
raised by the referees as discussed in detail below. We believe that the
feedback has helped us to further improve the manuscript.

In the discussion below, we have also addressed the weak points noted by the
two referees by stating a number of counter-arguments. We hope that in view of
these arguments and the amendments, our manuscript will now be found suitable
for publication in SciPost.

Best regards,
Tanja Schoger, Gert-Ludwig Ingold

Reply to referee 1

We thank the referee for the valuable comments and suggestions which
have led us to modify the manuscript as detailed below.

Firstly though, we would like to comment on the weakness listed by the referee.
While this point is certainly valid, we see a number of reasons why we think
that our paper deserves publication.

- While, as the referee points out and as we have stated already in the first
version of the manuscript, a combination of results in the literature
allows to obtain the main result of our paper, it has, to the best of our
knowledge, never been stated explicitly. As we had mentioned already in the
first version of the manuscript, statements in the literature indicate that
the existence of an analytical result for spheres with different radii was
not known in the Casimir community. Discussions with colleagues have supported
this view and we ourselves became aware of it only while finalising the
manuscript. As we discussed already in the first version, the high-temperature
result is relevant to the analysis of experiments even at lower temperatures
and therefore, we believe that it is important to explicitly state the result
(51) for the free energy.

- While our result could be obtained by combining results from three different
sources (refs. 5, 12, and 16), as far as we know it is the first time that
the Casimir free energy for two Drude spheres of different radii is derived
within a single framework, in our case within the scattering approach with
a plane-wave basis.

- The derivation of our main result may appear to be quite involved and it is
for this reason that we tried to be sufficiently explicit for the reader to
follow without too many difficulties. We like to highlight two more
formal points, though. Our calculation connects the scattering between two
different objects with an interesting combinatorial problem which is solved
on the way to our result (51). Furthermore, we expect that this approach might
also be useful in other problems related to the Casimir effect.

- For completeness, we mention the connection between Casimir physics and
electrostatics which our calculation highlights, a point which was also
acknowledged by the referee as a strength of our paper.

We now specifically discuss the referee's suggestions and how we addressed them.

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. [5] and [13], they will obtain the
formula using an alternative approach, providing the motivations for
doing that.


We have extended the introduction to explain in more detail the motivation of
our work. Specifically, we have modified the last part of the fourth paragraph
(previously the third paragraph) discussing the semi-analytical approach by
Bimonte (ref. 6) and the relevance of our work in this respect. We then have
modified the third paragraph and added two more paragraphs motivating our work
before discussing the structure of the paper.

2. In Section 4.4, the relation between Eq. (40) and the general result
of Ref. [13] should be clearly stated. The quantity Δ is formally evaluated
in [13] 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.


We have followed the advice and extended the discussion in the first part of
section 4.4 in order to make the connection between our result (41) (eq. 40
in the first version) and the paper by Fosco et al. [12] (ref. 13 in the first
version) even more explicit. Moreover, after (49), we now emphasize the
connection between the capacitance coefficients in [16] and our round-trip
description. Furthermore, we added the historic reference [27].

Reply to referee 2

First, we would like to thank the referee for the detailed and constructive
report. We are pleased that the referee finds our paper in principle suitable
for publication.

Before detailing the changes, we have made in response to the referee's
suggestions, thereby addressing point 2 of the listed weaknesses, we would like
to comment on point 1 where the referee points out that our result is only aimed
at a narrow audience.

While the length of the paper and the more formal character of parts of the
paper might indeed indicate that we address a narrow audience, we do not believe
this to be the case for the following reasons.

- The explicit result (51) for the Casimir free energy, from which the Casimir
force can be obtained, is not only relevant to theorists but also to
experimentalists while analysing Casimir experiments on two spheres with
different radii. We have rewritten the second part of the fourth paragraph
(previously the third paragraph) of the introduction to make this point clearer.

- While the plane-sphere geometry still is the most relevant geometry for
Casimir experiments, experiments on the sphere-sphere geometry have been
carried out lately. In particular, one of these experiments (ref. 7) addresses
larger distances, thereby rendering the contribution of thermal photons
and the zero-frequency Matsubara term more relevant. Furthermore, by
changing the salt concentration of the aequous medium, ref. 7 is able to
modify the strength of precisely the zero-frequency part. We have modified
the third paragraph to mention this point.

- Beyond the Casimir community in a more narrow sense, the sphere-sphere
geometry is relevant for the colloid community. To emphasize this point, we
have modified the first part of the third paragraph of the introduction by
splitting the earlier list of three experimental references into a part
pertaining to the Casimir effect (now refs. 7 and 8) and another part related
to colloidal systems (now ref. 9 and a new ref. 10). In colloidal systems, the
comment in the last two sentences of our previous point also applies in principle.

Therefore, we are convinced that our results are of interest not only to a small
group of theorists interested in a specific aspect of the Casimir effect but in
fact to a larger community.

As suggested by the referee, we have worked on the overall presentation and in
particular, addressed all points raised by the referee as detailed below. In a
couple of cases, we did not follow the referee for reasons given below as well.
We hope that the changes adequately address the referee's criticism.

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.


We significantly revised the first paragraph of our introduction to
make the role of thermal photons for the Casimir interaction clearer. In
particular, we discuss the equivalence of the classical limit (ℏ→0) and the
high-temperature limit (T→∞).

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


We addressed this point by the changes discussed in the previous point.

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


We wanted to convey, that the semi-analytical approach by Bimonte uses the derivative
expansion to determine the contributions from the non-zero Matsubara frequencies.
However, for the zero-frequency term, it is common to use the exact analytical expression.
The expressions are known for a sphere-plane geometry and a geometry of two equal spheres.
Our generalization of the classical result for a system of spheres with arbitrary radii
might therefore also be helpful for such semi-analytical approaches.

We hope that our revision of the relevant part of the introduction (the last part of
the fourth paragraph) clarifies our argument.

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 [5] but their result is given in a different form. I would thus say their
result is equivalent to that of [5] in this case.


Our result is dual, in the sense that the known expression [5] is expressed in
terms of bispherical multipoles. However, our approach is based on a round-trip
description. Using 'dual' instead of 'equivalent', we wanted to emphasise that
our result is not only equivalent to the known one but also allows for an
alternative interpretation in terms of round trips.

We have modified the sentence referring to section 3 in the last paragraph of
the introduction accordingly. More details concerning this duality are then given
later in the paper.

6 P3 Within the scattering approach to the Casimir effect [2], 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.


While we disagree that the free energy is always expressed as a sum over
Matsubara frequencies - one could also integrate over real frequencies - we
followed the suggestion of the referee and modified the text to avoid using
the term 'Matsubara sum'.

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.


We extended the first paragraph of section 2. Specifically, we now introduce the
dielectric function for a Drude metal and point out that the resulting finite dc
conductivity leads to a simplification concerning the modes contributing to
the Casimir free energy.

8 P5 Making use of the symmetries of cosine and hyperbolic cosine - I am not sure
what is meant by this


We were referring to the fact that adding π to the argument of the cosine will
merely change the sign and that the hyperbolic cosine is an even function. These
properties are of course well known and therefore we refrain from such a
detailed discussion in the paper. Since our original hint at the symmetries
apparently can constitute a source of confusion, we removed this sentence.

9 P5 again the same mysterious sentence - Casimir free energy which is found to
be dual to the known result [5].


While we added an explanation of the meaning of duality in the introduction, we
have replaced ‘dual’ by ‘equivalent’ in this case.

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 the authors mean the integrals in equation 8 which can be written as a
determinant ?


We do not think that it is necessary to amend the discussion following (9)
(previously eq. 8). We trust that the reader knows how to evaluate Gaussian
integrals. However, we want to emphasize that the non-trivial step in doing
these integrals is to evaluate the determinant of the bilinear form
in the exponent.

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


We agree with the statement in parentheses. Since a detailed discussion of
convergence properties is beside the main point of the paper, we removed the
term ‘considerably’.

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.


In our opinion, the additional explanations at the beginning of section 2 added
in response to the referee's point 7 are sufficient to understand the boundary

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 express more precisely.


We have extended the discussion at the beginning of section 4.1 accordingly. We
now recall the meaning of the -1 term in (9), previously eq. 8, to clarify how
each term accounting for at least one factor -1 in (9), contributes to the monopole
part for Dirichlet spheres.

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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