Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

Abstract

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.

Casimir Interaction between a Plane and a Sphere: Correction to the Proximity-Force Approximation at Intermediate Temperatures

“…At vanishing frequency, the reflection matrix elements of the sphere are diagonal with respect to polarization [34]. For the TM contribution, the plane-wave approach allows for the derivation of an exact analytic expression in the more general case of two spheres of arbitrary radii [70]. The previously known result for the plane-sphere geometry [59] is recovered as a particular case.…”

“…[60] by the multipolar approach. In Section 3.2, we focus on the TE zero-frequency contribution, as the TM correction can be more easily derived from an exact analytical representation obtained either by using bispherical coordinates [59] or by developing the plane-wave basis representation (7) [70].…”

Universal Casimir interactions in the sphere-sphere geometry

“…For the evaluation of the trace in (5), we adopt the plane-wave basis [28,29]. In view of the geometry of our problem, we decompose the wave vector into a transverse part k with modulus k and a zcomponent which after Wick rotation is written in terms of κ = [ε m (ξ/c) 2 + k 2 ] 1/2 .…”

“…In contrast to the scalar case discussed before, we do not have general expressions for the reduced free energy ( 15) and ( 16) for an arbitrary number of round-trips. In the case of two Drude spheres however, an exact expression is known for the sum over the contributions of all round-trips [29]. The difference with respect to the scalar case consists in the absence of the monopolar term associated with = 0 in the reflection matrix element (8) giving rise to the extra term −1 in (11).…”

“…The evaluation of the Gaussian integrals can be mapped onto a combinatorial problem of finding bicolored integer partitions as discussed in detail in Ref. 29. In the end, it turns out that the partitions are related to the elements of the dimensionless capacitance matrix of two spherical conductors.…”