# Impossibility of self constrained system of charged particles

1. Nov 30, 2009

### enroger0

Hi everyone, I remember reading there is a theoretical argument that said its absolutely impossible for systems of charged particles (plasma) to cobble together without external field. But I don't remember the name of the theorem.
Can anyone name it? Or even better point me to some nice materials on it? Thanks.

2. Nov 30, 2009

### Phrak

3. Nov 30, 2009

### enroger0

Thanks, I was able to track down the thing with the help of the paper. What I was looking for is Virial Theorem, am now looking at the wiki.

4. Dec 1, 2009

Staff Emeritus
That can't possibly be true. Here's a counterexample: a system of protons and electrons will "cobble together" to form hydrogen.

5. Dec 1, 2009

### enroger0

Yes, you are right. There must be some kind of assumption in the theorem, care to figure it out?

http://en.wikipedia.org/wiki/Virial_theorem

It's a general theory that deal with any force interaction, EM interaction takes special treatment.

6. Dec 1, 2009

### bcrowell

Staff Emeritus
The theorem that states that a system of EM particles can't be bound is a classical theorem; it doesn't apply to a qm system. I think the OP is mistaken about its being the same theorem as the virial theorem.

7. Dec 1, 2009

### enroger0

I was looking for the Virial Theorem, my motivation is along the line of why can't a hot plasma of any possible configuration confine itself by EM interaction between the particles alone. I think Virial Theorem answer that, but as Vanadium 50 point out: how about a pair of proton and electron? I don't know, still crunching math.

8. Dec 2, 2009

### bcrowell

Staff Emeritus
The theorem is classical. Classically, a hydrogen atom isn't stable. It collapses by radiation.

9. Dec 2, 2009

### Bob_for_short

If you speak of "hot" plasma - it has pressure inside and no pressure outside so it should expand.

If you speak of "cold" plasma, then it can be quasi-stable for some time but:

1) Thermal fluctuations lead to "vaporising" some particles and/or neutral clasters,
2) Atomic recombinations create neutral atoms that are not affected with binding forces and leave the plasma volume.
3) After recombination of all electrons and nuclei, you obtain a hot gas with pressure which should expand anyway.

In other words, plasma is a too energetic system to be in a "condensed" state.

Last edited: Dec 2, 2009
10. Dec 2, 2009

### arivero

Well if we forget about looking for general theorems and constrain ourselves to electromagnetic charged classical particles, it seems easy. As Bcrowell has said in #8, A non static system will radiate and loss energy. We are then looking for static solutions.

11. Dec 2, 2009

### Bob_for_short

If we look at the Universe, most of the matter is in a plasma state. I think a cold and rare plasma is rather stable.

12. Dec 2, 2009

### arivero

Bob, forget gravity.

13. Dec 2, 2009

### arivero

Perhaps it is marginally related to this topic... I think to remember that classical thermodynamics does not apply to self-graviting bodies, but I can not remember the precise statement. Something about existence of KMS states?

14. Dec 2, 2009

### Bob_for_short

I did no mean gravity (stars) as an attractive force. I spoke of far inter-stellar space. Sorry, I was not clear.

15. Dec 2, 2009

### bcrowell

Staff Emeritus
Right. I know the theorem exists, but I don't know the exact statement of the theorem, and I don't know exactly what's involved in the proof. You would have to worry about things like cases where the particles are moving without accelerating, so that they don't radiate. You could also have cases where the particles are accelerating, but they cancel out each other's radiation, etc. And I'm not sure what the theorem actually claims happens. I think it probably states that, e.g., the system decomposes into pointlike, electrically neutral pieces, which then fly off without any further interactions.

16. Dec 2, 2009

### Bob_for_short

If we speak of hot plasma, there are two regimes of its radiation: from its volume and from its surface. When the plasma is dense and thick, it reabsorbs its own volume radiation so it radiates only from its surface (black body radiation determined with T).

It is very important in laboratory conditions to pass quickly the first regime while ionizing the neutral gas in order to decrease the radiative losses from volume. It's a question to be transparent or non transparent for radiation. But a hot plasma has pressure and a lot of instabilities even when confined.

Last edited: Dec 2, 2009
17. Dec 2, 2009

### bcrowell

Staff Emeritus
There seems to be a family of theorems of this general flavor.

Earnshaw's theorem: http://en.wikipedia.org/wiki/Earnshaw's_theorem , http://www.everything2.com/index.pl?node_id=1449752 http://math.ucr.edu/home/baez/physics/General/Levitation/levitation.html

optical Earnshaw's theorem: http://www.everything2.com/title/magneto-optical%20trap

Some of the links also discuss the magnetic versions of the theorem.

The form of the theorem that I'm think of doesn't seem to fit with any of these. I seem to recall that Prof. Rich Muller told my Physics H7 class at Berkeley about such a theorem ca. 1983, saying that someone he knew (possibly Purcell, the author of the text we were using?) had proved it, but hadn't published the proof.

Last edited by a moderator: Apr 24, 2017
18. Dec 2, 2009

### enroger0

Hot plasma have pressure, if it's just a ordinary blob of hot gas then it obviously will expand. But there remains a remote possibility that the plasma having an internal current inducing B field to somehow contain itself (scifi plasma bolt anyone?). Radiation lost and thermodynamic lost (particle collision) will kill the configuration eventually, but it could be quasi-stable.

19. Dec 4, 2009

### takeTwo

Our solar system is many-bodied. When it formed, it had billions more bodies. Yet we are just now figuring out how *neutral* blobs of rock and gas came together to form the planets beyond Jupiter. Now, add EM fields. Seems rather untraceable for a collections of particles we might call a "plasma"; assuming that is what is meant by a "collection of charged particles". Whether it is gravity, EM fields, or lining up your cue stick to break the rack of balls in pool; if there are enough degrees of freedom, the problem is hard. Impossible for some self-assembly? No idea. Nature is much more creative than myself!

If there are enough electrons or ions such that the number of them in a "Debye sphere" = { 740 * SQRT[ T(degrees K) / n(cm^-3) ] }^3 >1, then collective phenomena (non-local EM forces) occur and thus the problem with fusion energy!! The components of the hydrogen atom fail this test of "plasma or not". Systems of not-plasma must be common, say, in interstellar space (as Bob for short mentioned), but in laboratories, I can't think of any (maybe electron traps, but they are not self-constrained).

However, there is "ball lightning". This is the one (supposedly) self-constrained system of charged particles that I've ever hear of.

20. Dec 4, 2009

### arivero

So at the end the question is about the stability of "ball lightning". Uff.

21. Dec 4, 2009

### enroger0

crap, you caught me.

22. Dec 4, 2009

### Bob_for_short

AFAIK, the ball lightning is not really stable but actively decaying. Some chemical reaction like fire or so.

23. Dec 4, 2009

### bcrowell

Staff Emeritus
We're talking about specific mathematical theorems here. For instance, Earnshaw's theorem refers to classical systems in stable, static equilibrium. The solar system is a nonclassical system (the atoms have nonclassical properties) that is not in static equilibrium.

24. Dec 4, 2009

### takeTwo

OK. I was just thinking out loud. I like analogies: here is another many-bodied, classical system. In any case, doesn't the theorem refer to the lack of stable, static equilibria for classical system under 1/r^2 forces such as coulomb forces and gravitational forces?

The long-range gravitational forces of the (classical) clumps of rocks (not molecules) of our solar system during, say, only the last millions of years is still not in any kind of equilibrium. Doesn't this agree with Earnshaw's theorem? Why did comet Shoemaker-Levy collide with Jupiter? Whatever perturbation it experienced sent it into a different orbit; the 1/r^2 forces are not restorative.