Around a theorem of F. Dyson and A. Lenard: Energy equilibria for point charge distributions in classical electrostatics

Dedicated to the memory of Freeman Dyson (1923–2020)
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Abstract

We discuss several results in electrostatics: Onsager’s inequality, an extension of Earnshaw’s theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.

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 x,yRd,d3, we define the function Kd(x,y) by the formula Kd(x,y)=1(2d)ωd11|xy|d2.Here, ωd1 is the surface area of the unit sphere in Rd. Kd is the fundamental solution for the Laplace operator in Rd (i.e., ΔyKd(x,y)=δx).

Furthermore, given a finite (signed) Borel measure μ with support Σ, we define the Newtonian (or Coulomb) potential of μ with respect to the kernel Kd(x,y) by UΣμ(x)=ΣKd(x,y)dμ(y).If the support of the measure μ is either clear from context or, otherwise, irrelevant to the problem, we drop the subscript Σ, and write Uμ(x).

We define the Newtonian (or Coulomb) energy of a measure μ (with no atoms) with respect to the kernel Kd(x,y) by WΣ[μ]=ΣUΣμ(x)dμ(x).If μ 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 μ=kQkδxk, then Σ={xk}k=1n, and the energy is set to be WΣ[μ]=1kjnQjQkKd(xj,xk),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 WΣ[μ]=W[μ] — cf. [9], [13] for the basics of potential theory.

Definition 1

Let m0, {Σj}j=1m, Σ=j=1mΣj be a collection of compact sets, and {Qj}j=1m a collection of charges. We say a charge distribution (measure) μ, supported on Σ, is in constrained equilibrium, when its electrostatic potential UΣμ(x)=ΣKd(x,y)dμ(y),is constant (possibly taking different values) on each connected component Σj of the support of μ, subject to the constraints μj=μ(Σj)=Qj,j=1,2,3,,m.We also define the quantity Q=j=1mQj, which we refer to as the total charge of the system.

In comparison, for the classical equilibrium problem of minimizing the energy WΣ[μ], there is a unique minimizing measure whose potential is constant (quasi)-everywhere on the support of μ; the notion of constrained equilibrium is thus a generalization of the usual equilibrium problem, with the potential being locally constant on the support of μ.

In the case when the Σj’s consist only of a single point, i.e. Σj={xj}, the condition that the potential UΣμ be constant is replaced by the gradient condition UΣΣjμ(xj)=0;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 QR and a finite Borel measure μ, such that μ(Rn)=Q, is there a collection of disjoint compact sets {Σj}j=1m, such that μ(Σj)=Qj, j=1mQj=Q, and μ has a constrained equilibrium configuration on ΣjΣj?

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 Σ and set of constraints (1.6).

Remark 1

It should also be noted that the choice of total charge Q is not important, except to distinguish between neutral configurations (Q=0) and non-neutral (Q0). This is due to the fact that, under a simple rescaling Q=λq,Qj=λqj,λR{0},the potential scales by a factor of λ, and the energy by a factor of λ2, leaving the variational problems (and their solutions) unchanged. Therefore, the only two distinct cases that need to be considered are Q=0 and Q=1.

The second (classical) problem we discuss is the characterization of critical manifolds C (specifically, curves) on which the gradient of the potential of a given charge configuration vanishes. More precisely, the problem in R2 is:

Problem 2

Let μ be a finite charge distribution with planar support ΣR2. Can the critical manifold CR3 of μ, defined by C={xR3:UΣμ(x)=0}contain a curve in R2?

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 μ, 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 n point charges {xj}j=1n are placed in R3, then there are at most (n1)2 points in R3 at which the electrostatic field vanishes. More precisely, if each xj has charge qjR, then #{x:(j=1nqj|xxj|)=0}(n1)2,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 n=3, the best known estimate is 12, not 4! 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 (n1) 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 R3), 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 Rn.

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 R2). 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 Rn, n>2.

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 R3. 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 μ in R3 with support contained in a plane HR2 (without loss of generality, we take H to be the xy-plane). Then the critical manifold CR3 of μ, defined by C={xR3:Uμ(x)=0}cannot contain a curve in H.

In other words, if C contains a curve on which Uμ(x)=0, then the latter is necessarily transversal to the plane H.

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 B={xRd:|x|<1} be the unit ball in Rd, d>2, and μ be a (signed) charge distribution supported on the closure of B which produces the same electrostatic potential outside of B¯ 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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The author’s research is supported by the Simons Foundation, United States of America , under the grant 513381.

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