Instability of classical system of point charges

[bcrowell]

Earnshaw's theorem http://en.wikipedia.org/wiki/Earnshaw's_theorem gives a straightforward reason why we can't have a static equilibrium for a system of point charges. For some time I've been trying to find out if anyone's worked out a similar proof for the impossibility of a dynamic equilibrium. I had a memory of having heard such a theorem stated by one of my undergrad profs ca. 1983, and I went so far as to contact him and ask, but he didn't remember anything. Does anyone know anything about this topic?

The plausibility of such a result comes from the fact that by Earnshaw's theorem the charges must accelerate, and accelerating charges usually radiate. Therefore we would expect an arrangement of charges in some localized region of space to steadily lose energy and start to collapse. The end result would probably be either a single pointlike object or possibly multiple such objects heading off to infinity relative to one another. It's not obvious to me what is the best way, in general, to define stability or instability, so a more precise way to frame this is part of the problem. Maybe one way to define a dynamical equilibrium is that both the positions and the velocities of the particles remain bounded in magntiude. Or maybe there is some way to pose this in terms of statistical mechanics; collapse or breakup clearly increases the system's entropy.

The difficulty in proving such a result formally is that acceleration is a necessary but not sufficient condition for radiation. For example, if a spinning, uniformly charged, insulating sphere has inertial center-of-mass motion, it will not radiate, even though the elements of charge are all accelerating. There are even some examples that can be constructed of rigid charge distributions (in some cases asymmetric ones) that spin and undergo center-of-mass acceleration without radiating.[Goedeke 1964]

Goedeke speculates that "if we want the radiation to be exactly zero ... we probably cannot built a distribution out of point charges," although it's not obvious to me whether he means this statement to apply only to his rigid-rotation examples, and in any case he's only forming a conjecture. But it seems reasonable that the conjecture I'm concerned with should only be posed for a finite number of particles. (And I'm only talking about purely electromagnetic systems of point charges, whereas Goedeke's systems need other forces to maintain their rigidity.)

It's also interesting to think about how to pose and prove such a theorem in GR for a gravitationally interacting system, but it seems much tougher since you don't have a background spacetime.

By the way, googling about this stuff will turn up a lot of material by and about a kook named Randell Mills, who claims to have disproved quantum mechanics and found an unlimited source of energy. There are a couple of WP articles, "Nonradiation condition" and "BlackLight Power," where he has tried to promote himself. I think the former misstates the results by Goedeke and Haus, as well as vastly overstating their importance.

Goedeke 1964, http://journals.aps.org/pr/abstract/10.1103/PhysRev.135.B281 ; pdf at https://web.archive.org/web/20010305011426/http://www.hydrino.org/Papers/Goedecke.pdf

 

[mathman]

When point charges get close enough quantum theory takes over. Since they have to be of opposite sign, there are essentially two cases to consider. (1) Electron plus proton - results in Hydrogen atom. (2) Electron plus positron - results in annihilation.

 

[bcrowell]

When point charges get close enough quantum theory takes over. Since they have to be of opposite sign, there are essentially two cases to consider. (1) Electron plus proton - results in Hydrogen atom. (2) Electron plus positron - results in annihilation.

I'm talking about the classical system.

 

[Jano L.]

Earnshaw's theorem http://en.wikipedia.org/wiki/Earnshaw's_theorem gives a straightforward reason why we can't have a static equilibrium for a system of point charges. For some time I've been trying to find out if anyone's worked out a similar proof for the impossibility of a dynamic equilibrium. I had a memory of having heard such a theorem stated by one of my undergrad profs ca. 1983, and I went so far as to contact him and ask, but he didn't remember anything. Does anyone know anything about this topic?

For a model with only retarded fields, stability of a system of two oppositely charged particles interacting exclusively via EM forces seems very unlikely. There is a paper by John Synge, which reports on a numerical calculation of evolution of such a two-body system:

J. L. Synge, On the Electromagnetic Two-Body Problem, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 177, No. 968 (Dec. 31, 1940), pp. 118-139

http://www.jstor.org/stable/97589?seq=1#page_scan_tab_contents

His conclusion is that the particles approach each other, so the system is unstable. I think his conclusion is right, because retarded fields have outward directionality and thus allow for leak of EM energy to infinity.

When other EM fields are considered, the dynamics of the system may be very different.

For example, Page and Frenkel have investigated half-retarded, half-advanced field models:

L. Page, Advanced Potentials and their Application to Atomic Models, Phys. Rev. 24, 296 (1924)
http://dx.doi.org/10.1103/PhysRev.24.296

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
http://dx.doi.org/10.1007/BF01331692

Such models seem to allow for stable motions.

Another option to assume retarded interactions like in Synge's calculation, but assume particles experience additional "background noise" field. The noise may prevent the collapse Synge has described. Cole and Zhou have published results of numerical calculations that support this idea:

D. C. Cole and Y. Zou, “Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics,” Physics Letters A, Vol. 317, No. 1-2, pp. 14-20, 13 October 2003.
http://dx.doi.org/10.1016/j.physleta.2003.08.022

 

[bcrowell]

Interesting stuff, Jano, thanks!

This is good for helping me to crystallize a rigorous statement of the problem. I have in mind a system that is isolated in infinite space, not a system in a box as in the Cole paper. (I don't buy Cole's apparent belief that the box is just a computational convenience, and in general I think the paper is a little crankish.) I also have in mind a system that is free to radiate to null infinity but that is not receiving any radiation originating from the outside.

I agree that a violation of the conjecture seems highly unlikely, but that's not the same as a rigorous proof. The Goedeke paper, for example, shows that some highly intuitive things can happen. Even with a finite system of point particles, very weird possibilities exist. An outrageous example with Newtonian gravity is Z. Xia, “The Existence of Noncollision Singularities in Newtonian Systems,” Annals Math. 135, 411-468, 1992, described at https://plus.maths.org/content/outer-space-twos-company-threes-crowd .

 

[Jano L.]

Interesting stuff, Jano, thanks!

This is good for helping me to crystallize a rigorous statement of the problem. I have in mind a system that is isolated in infinite space, not a system in a box as in the Cole paper. (I don't buy Cole's apparent belief that the box is just a computational convenience, and in general I think the paper is a little crankish.)

You're welcome.
Using a box is a standard way to make Fourier analysis computationally feasible. Where do you see the problem? Give some arguments, not opinions please.

 

[bcrowell]

You're welcome.
Using a box is a standard way to make Fourier analysis computationally feasible. Where do you see the problem? Give some arguments, not opinions please.

They seem to believe that their computation demonstrates the possibility that quantum mechanics is somehow emergent from classical electromagnetism. This idea has been explored before by people like Schott (1933) and Goedeke (1964). In all of these attempts, you have to find some way to explain how Planck's constant emerges, since classical E&M doesn't contain any fundamental quantity with the correct units. In the older attempts, it emerges from the properties of the subatomic particle, which is modeled as a rigid body of finite size, presumably held together by some unknown force. In the Cole paper, the particles are treated as pointlike, and therefore Planck's constant can't emerge from their assumed size or structure. The only place I can see that it can come from is from some other dimensionful quantity that was inserted into the calculations, which AFAICT would have to be the size of the box or the cutoff in the wavelengths. So when they display probability distributions for their electrons, and remark on how wonderful it is that these resemble the quantum-mechanical ones, we have to ask what sets the scale for these distributions. (Actually I couldn't find the missing figures on arxiv, where the paper claims they can be found.)

Even if you put aside the crankishness of the claims about QM as emergent from E&M, the existence of the background of electromagnetic waves is clearly an artifact of the box. If there had been no box, the waves would have radiated away to infinity. This would continuously suck energy out of the system and eliminate its dynamical stability.

 

[Jano L.]

In all of these attempts, you have to find some way to explain how Planck's constant emerges, since classical E&M doesn't contain any fundamental quantity with the correct units. In the Cole paper, the particles are treated as pointlike, and therefore Planck's constant can't emerge from their assumed size or structure.

I do not think that is a reasonable requirement on their work. Value of Planck's constant was originally obtained by fitting the Planck function to the measurements of thermal spectrum. It was never explained how it emerges from something else in quantum theory. Why should it be necessary to explain it by classical theory? The goal of the authors is elsewhere. They work within the scheme of stochastic electrodynamics, where the Planck constant enters through the characteristic function usually called "energy spectral function" of the background fields.

So when they display probability distributions for their electrons, and remark on how wonderful it is that these resemble the quantum-mechanical ones, we have to ask what sets the scale for these distributions.

From the paper:

The random number generator algorithm was designed to produce a Gaussian distribution for these coefficients, with an expectation value of zero, and a second moment of, $\langle A_{\mathbf k_n,\lambda}\rangle = \langle B_{\mathbf k_n,\lambda}\rangle = 2\pi\hbar \omega_{n}$. The latter specification corresponds to the energy spectrum of classical electromagnetic ZP radiation of $\rho_{ZP}(\omega) = \hbar \omega^3/(2\pi c^3)$[6].

So the answer to your question is: the characteristic function of the background noise $\rho_{ZP}$ that they use to formulate their model and numerical scheme.

... the existence of the background of electromagnetic waves is clearly an artifact of the box. If there had been no box, the waves would have radiated away to infinity. This would continuously suck energy out of the system and eliminate its dynamical stability.

Unless there is background radiation everywhere. Do you think existence of cosmic background radiation is an artifact of some box?
No, the box is just a standard requisite for using the Fourier series to express functions of three coordinates.

 

[vjacheslav]

Who is Earnshow, actually, and what are the reasons to believe his theorem?

 

[bcrowell]

@Jano L. -- Clearly we disagree on the interpretation of the Cole paper, but I don't think that has any relevance to the topic of this thread.

Who is Earnshow, actually, and what are the reasons to believe his theorem?

https://en.wikipedia.org/wiki/Earnshaw's_theorem

 

[Jano L.]

@Jano L. -- Clearly we disagree on the interpretation of the Cole paper, but I don't think that has any relevance to the topic of this thread.

Let's agree to disagree, then.

 

[bcrowell]

It occurs to me that there may be a reason that this problem is not clearly defined. It's formulated in terms of pointlike particles, and a common, expected type of instability is that many of these particles will end up at the same location at the same time. This is what Synge refers to in the 1940 paper as "degeneracy." (This kind of degeneracy is a cataclysmic event, since it releases an infinite amount of energy.) But the classical equations of motion probably lack self-consistency, because when the size of of our particle is small compared to the classical electron radius, the mass equivalent of the electrostatic energy is greater than the mass of the particle. Synge insists that the model he uses *is* self-consistent, but says that it violates conservation of energy, which seems nearly as bad.

A related issue is that regardless of whether you make your particles pointlike or not, you can't give the initial conditions as positions, velocities, and fields on a Cauchy surface. You need more information. For example, if you consider an extended body, the initial conditions need to give the motion over a time equal to the size of the particle over c: Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A39 (2006) 3801-3816, http://arxiv.org/abs/physics/0508031 . Medina's interpretation is that this is because bodies can't be perfectly rigid, and this is the time required for internal oscillations to die down. If you make the particles pointlike, you need to give information such as the initial acceleration.

 

[Jano L.]

But the classical equations of motion probably lack self-consistency, because when the size of of our particle is small compared to the classical electron radius, the mass equivalent of the electrostatic energy is greater than the mass of the particle.

Which equations do you mean? Synge considers point particles with given mass. There is no electrostatic mass in his model.

Synge insists that the model he uses *is* self-consistent, but says that it violates conservation of energy, which seems nearly as bad.

I am sure his model is consistent (about his numerical scheme I do not know). Whether the system conserves energy depends on which kinds of energy are admitted. Synge probably meant that sum of kinetic energies of the particles and their Coulomb potential energy is not conserved, which is true. However, if energy of field is taken into consideration, the local conservation of energy holds; any energy lost from the system can be expressed as integral over surface enclosing the system.

A related issue is that regardless of whether you make your particles pointlike or not, you can't give the initial conditions as positions, velocities, and fields on a Cauchy surface. You need more information. For example, if you consider an extended body, the initial conditions need to give the motion over a time equal to the size of the particle over c: Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A39 (2006) 3801-3816,

http://arxiv.org/abs/physics/0508031 . Medina's interpretation is that this is because bodies can't be perfectly rigid, and this is the time required for internal oscillations to die down. If you make the particles pointlike, you need to give information such as the initial acceleration.

I am not sure about the extended particles, but for point particles, giving positions and velocities of point particles and all the fields is sufficient to integrate both the equations of motion for particles and time-dependent equations for field. Of course, choosing definite values of all the fields at all points of space is problematic in general, there are many possibilities. One can calculate retarded fields of a prescribed motion and then use those as an initial condition for subsequent integration.

 

[bcrowell]

Which equations do you mean?

See pp. 119 and 121 of the Synge paper.

Synge considers point particles with given mass. There is no electrostatic mass in his model.

Synge's treatment is relativistic. In a self-consistent relativistic treatment, you can't ignore the mass-energy due to the electric field, and that mass-energy diverges, which causes problems. A good reference on this topic is Poisson, http://arxiv.org/abs/gr-qc/9912045 .

I am sure his model is consistent (about his numerical scheme I do not know).

It's an analytic calculation, not a numerical one.

Whether the system conserves energy depends on which kinds of energy are admitted.Synge probably meant that sum of kinetic energies of the particles and their Coulomb potential energy is not conserved, which is true.

No, I don't think that's what he means.

I am not sure about the extended particles, but for point particles, giving positions and velocities of point particles and all the fields is sufficient to integrate both the equations of motion for particles and time-dependent equations for field.

Not true. See the Poisson paper, and also Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A39 (2006) 3801-3816, http://arxiv.org/abs/physics/0508031 .

 

[Jano L.]

See pp. 119 and 121 of the Synge paper.

Synge knew there is no self-consistency problem with his assumptions:

It is perhaps necessary to insist on their logical consistency, because there appears to be an opinion that it is necessary to supplement the equations of motion with a radiation term, involving the rate of change of acceleration. The arguments in this connexion do not show any intrinsic inconsistency between these hyptheses and the law of conservation of energy, assuming that the energy in the field may be represented by the usual form of energy-tensor.
It is hoped in a later paper to discuss this question of the conservation of energy with some care, in connexion with a modified definition of the stress-energy tensor which avoids any inconsistency between the three hypotheses stated above and the law of conservation.

I do not know whether he published that paper, but the expression of the appropriate stress-energy tensor can be found in other papers working with these assumptions, e.g.

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. | http://dx.doi.org/10.1007/BF01331692J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct
Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.
| http://dx.doi.org/10.1103/RevModPhys.21.425R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Letters, 8, 3, (1964), p. 185-187. | http://dx.doi.org/10.1016/S0031-9163(64)91989-4

If you'll read these carefully, I am certain you'll come to conclusion there is no obvious self-consistency problem there.

Synge's treatment is relativistic. In a self-consistent relativistic treatment, you can't ignore the mass-energy due to the electric field, and that mass-energy diverges, which causes problems. A good reference on this topic is Poisson, http://arxiv.org/abs/gr-qc/9912045 .

...

See the Poisson paper, and also Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A39 (2006) 3801-3816, http://arxiv.org/abs/physics/0508031 .

Those problems only occur in theories that make assumption field energy is given by the Poynting formulae. For point particles, this assumption is what is invalid in the first place. If used anyway, this leads to unphysical dynamics and contradictions.

Both Poisson and Medina in their papers seem to fall into that old trap thinking point particles are limit of extended particles. That indeed does lead to problems.

Fortunately, one can work with point particles without relying on the old continuum EM theory and its Poynting energy and Maxwell stress formulae. Above papers show that point particles allow for consistent work-energy theorem, with a different formulae for EM energy and EM stress tensor.

 

[bcrowell]

There seem to be some results in GR that are sort of similar to what I'm talking about, e.g., this: Lindblad and Rodnianski, "The global stability of the Minkowski space-time in harmonic gauge," http://arxiv.org/abs/math/0411109 . This is for a scalar field. The paper is pretty technical, and I'm not clear on where they ever define the crucial word "converging" in the statement of the problem near the bottom of p. 2.

 

[bcrowell]

Also relevant:

https://www.physicsforums.com/threads/struggles-with-the-continuum-part-3-comments.832462/

Eliezer, 1943, https://googledrive.com/host/0B8_joYJa2eNzfkNIQ2ZiY1hxS0FlMTI4ZC1IYnRUZ0ttd0NrQlYxcEdMZnFSMnpYZ3ltbUE/v39/3/S0305004100017850.$