Around a theorem of F. Dyson and A. Lenard: Energy equilibria for point charge distributions in classical electrostatics
Introduction
Electrostatics is an ancient subject, as far as most mathematicians and physicists are concerned. After several centuries of meticulous study by people like Gauss, Faraday, and Maxwell (to name a few), one wonders if it is still possible to find surprising or new results in the field. Throughout this text, we address several seemingly classical electrostatics problems that have not been fully addressed in the literature, to the best of our knowledge. Let us begin by establishing some notations.
For vectors , we define the function by the formula Here, is the surface area of the unit sphere in . is the fundamental solution for the Laplace operator in (i.e., ).
Furthermore, given a finite (signed) Borel measure with support , we define the Newtonian (or Coulomb) potential of with respect to the kernel by If the support of the measure is either clear from context or, otherwise, irrelevant to the problem, we drop the subscript , and write .
We define the Newtonian (or Coulomb) energy of a measure (with no atoms) with respect to the kernel by If has atoms, i.e., point charges, the definition above is modified slightly to guarantee finiteness of the energy, in accordance with the usual physics convention. If , then , and the energy is set to be which coincides with the previous definition if all integrals are taken in the principal value sense. We also refer to this functional as the electrostatic energy, and sometimes use the simpler notation — cf. [9], [13] for the basics of potential theory.
Definition 1 Let , , be a collection of compact sets, and a collection of charges. We say a charge distribution (measure) , supported on , is in constrained equilibrium, when its electrostatic potential is constant (possibly taking different values) on each connected component of the support of , subject to the constraints We also define the quantity , which we refer to as the total charge of the system.
In comparison, for the classical equilibrium problem of minimizing the energy , there is a unique minimizing measure whose potential is constant (quasi)-everywhere on the support of ; the notion of constrained equilibrium is thus a generalization of the usual equilibrium problem, with the potential being locally constant on the support of .
In the case when the ’s consist only of a single point, i.e. , the condition that the potential be constant is replaced by the gradient condition This is the appropriate analog for systems of point charges.
Now, we are ready to state the first problem discussed in this paper.
Problem 1 Given a total charge and a finite Borel measure , such that , is there a collection of disjoint compact sets , such that , , and has a constrained equilibrium configuration on ?
A solution for Problem 1 merely gives an equilibrium charge configuration, which need not be also a stable equilibrium, i.e. a local minimum of the energy functional, as opposed to a saddle point. The (stronger) stable equilibrium problem may not have a solution, for a generic choice of the support and set of constraints (1.6).
Remark 1 It should also be noted that the choice of total charge is not important, except to distinguish between neutral configurations () and non-neutral (). This is due to the fact that, under a simple rescaling the potential scales by a factor of , and the energy by a factor of , leaving the variational problems (and their solutions) unchanged. Therefore, the only two distinct cases that need to be considered are and .
The second (classical) problem we discuss is the characterization of critical manifolds (specifically, curves) on which the gradient of the potential of a given charge configuration vanishes. More precisely, the problem in is:
Problem 2 Let be a finite charge distribution with planar support . Can the critical manifold of , defined by contain a curve in ?
This problem has numerous applications, some of which are discussed in this paper. One of its most obvious implications is that a collection of charges, placed in the plane supporting the distribution , cannot have a curve in the same plane as an equilibrium configuration.
Problem 2 has a distinguished history, and can be regarded as a special degenerate case of Maxwell’s Problem (cf. [6] for further references). In short, Maxwell asserted (without proof) that if point charges are placed in , then there are at most points in at which the electrostatic field vanishes. More precisely, if each has charge , then or is, otherwise, infinite (e.g. contains a curve, the ‘degenerate case’, which is partially addressed in Problem 2). Thompson, while preparing Maxwell’s works for publication, could not prove it, and the problem has since become known as Maxwell’s conjecture — cf. [6] for more details. So far, the only real progress on this problem has been achieved in [6], where it was verified that the cardinality of the set of isolated points in equilibrium in (1.9) is finite. But even in the case of , the best known estimate is , not ! No counterexamples to the conjecture have been found.
The first examples of curves of degeneracy where (1.9) holds go back to A.I. Janušauskas [7].2 Further, partial results, for a particular case when the charges are coplanar can be found in [12], [17]. Killian, in particular, conjectured that in the latter case the degeneracy curves are all transversal to the plane of the charges. This is proven in Section 4. Finally, note that in the plane, if we use the logarithmic potential, the estimate improves to and is an obvious corollary of the Fundamental Theorem of Algebra.
This paper is organized as follows: in Section 2, we discuss a particular bound for the electrostatic energy of a system of point charges in geometric terms, first in its classical form (for the Newtonian potential in ), and the proof of F. Dyson and A. Lenard for an inequality first discovered by L. Onsager [16], along with extensions of the same energy inequality to the case of potentials in .
In Section 3 we focus on necessary conditions for the existence of an equilibrium configuration, as in Problem 1, in particular for the case of Coulomb potentials (in ). A necessary condition independent of the support, and which can be expressed as a constraint on the measure density moments, is also discussed in Section 3 (Intersection Theorem). We also discuss some generalizations of this theorem to , .
Section 4 is dedicated to the precise formulation and solution of degeneracy in Maxwell’s problem, for charge configurations constrained to two-dimensional subspaces of . In Section 5 we pose a fascinating question, originating in approximation theory, that we frivolously label ‘Faraday’s problem’, believing that Sir Michael Faraday would have never hesitated to answer it based on empirical evidence.
Section snippets
Variations on Onsager’s inequality
The Onsager inequality was originally discussed by Lars Onsager3 in a relatively little known paper [16]. Onsager himself did not provide a proof of this inequality, and it was not
Intersection theorem
As we have already seen, two-dimensional electrostatics is rather special; the following theorem is no exception to this rule. The theorem may have been known earlier, and the authors would be interested to see if one could find the earliest instance of it in the literature.5 We remark that it is a
Degeneracy in Maxwell’s problem with planar charge distributions
We now address Problem 2, which stems from Maxwell’s conjecture. The following question was first discussed in [12], also cf. [17]:
Proposition 4.1 Consider a distribution of point charges in with support contained in a plane (without loss of generality, we take to be the -plane). Then the critical manifold of , defined by cannot contain a curve in .
In other words, if contains a curve on which , then the latter is necessarily transversal to the plane .
Proof To see
Faraday’s problem
We conclude this exposition with the following question, which came up in connection with the seemingly unrelated problem of uniqueness of the best uniform approximation by harmonic functions from approximation theory [11]. However, the problem is, in spirit, very close to the subject of this paper. Let be the unit ball in , , and be a (signed) charge distribution supported on the closure of which produces the same electrostatic potential outside of as the point
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
We would like to thank the referee, whose comments helped to clarify and improve the exposition.
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