Electron degeneracy pressure in QM

[Naty1]

In BLACK HOLES AND TIME WARPS by Kip Thorne there is a fascinating discussion about how electron degeneracy pressure fights gravity in dense stars, beginning on page 146. He says the following (abbreviated excerpts) :

"..Quantum mechanics insists that when already dense matter is compressed a bit...the energy of the degeneracy motion must increase...If the electon is already moving at close to the speed of light...the additional energy goes into inertia...these two different fates of energy (added speed vs added inertia) produce different increases in the electron's pressure and thus different resistances to compression."

In effect this means that a star's resistance to gravitational collapse decreases with increasing density, not what intuition might suggest.

[Without this difference, if I understood the discussion, we'd not have any black holes.]

My question is why this difference in pressure happens...do we have a physical interpretation...why electron velocity has a different effect on pressure than inertial change??


Apparently the general relationship of white dwarf matter density is governed by the Stoner-Anderson equation of state which shows that..."matters resistance to compression decreases smoothly from 5/3 to 4/3...as electrons speed into the relativistic domain.."

and within the equation might be the answer to my question above...I did not find any Wikipedia entry for Stoner-Anderson...

Thanks

 

[Matterwave]

Relativistic electrons cannot exert the same degeneracy pressure as non-relativistic electrons. There are several derivations of this using either E=p^2/2m+mc^2 or E^2=p^2c^2+m^2c^4.
The decreased ability to keep up degeneracy pressure comes out in the math, but I've never really heard of a conceptual answer...

 

[bcrowell]

Here is my own explanation of what Thorne is talking about (see subsection 4.4.3): http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.4

Essentially the idea is that one factor in determining the pressure exerted by a gas is the time it takes a particle to rattle back and forth in the space it's confined in. If this time is short, then there are more impacts, and more pressure. For a degenerate gas, energy gets higher as you increase the density. (This is just the Heisenberg uncertainty principle.) Once the speeds become relativistic, you can no longer decrease this time between impacts by adding energy. The dependence of pressure on density therefore becomes less drastic for a highly relativistic degenerate gas ($P \propto \rho^{4/3}$) than for a nonrelativistic one ($P \propto \rho^{5/3}$).

 

[Ich]

bcrowell,

I did a quick web search yesterday to refresh my memory, but your explanation is far better than everything else I found.

 

[Altabeh]

Once the speeds become relativistic, you can no longer decrease this time between impacts by adding energy. The dependence of pressure on density therefore becomes less drastic for a highly relativistic degenerate gas ($P \propto \rho^{4/3}$) than for a nonrelativistic one ($P \propto \rho^{5/3}$).

But to my own knowledge, the more energy you add to any material system, the faster its particles oscillate so the more impacts will happen. This has nothing to do with relativistic speeds as far as the total energy of the system amounts to a certain binding energy.

Maybe relativity here makes the particles go beyond this energy and then impacts happening in the system decrease because particles are no longer bound to that system due to its collapse! Can you discuss this?

AB

 

[Ich]

But to my own knowledge, the more energy you add to any material system, the faster its particles oscillate so the more impacts will happen.

This is not about oscillations, and not even about impacts. It's about (symmetric) momentum transport across a surface, which is proportional to the number of particles crossing per time unit, which in turn is proportional to the particle velocity, which is bounded.

 

[Altabeh]

This is not about oscillations, and not even about impacts. It's about (symmetric) momentum transport across a surface, which is proportional to the number of particles crossing per time unit, which in turn is proportional to the particle velocity, which is bounded.

I just shared my own knowledge of how the number of impacts in a mateial system varies by adding energy into it which was seemed to be converse from bcrowell's viewpoint! Even if we assume that you've not misunderstood me, the number of particles crossing through the surface (of material system) per time unit does strongly depend on the energy stored in the system. The velocity of particle is bounded but if the energy injected into the system gets higher, this velocity is no longer limited unless we say that conservation law of energy and momentum do not hold!

As stars generates energy by nuclear fusion, there must be heat produced from the reactions of fusion which in turn prevents gravitational collapse of the star. But as the heat carries energy, so the energy of system increases, giving rise to the increase of the number of particles crossing through the surface of star. With this happening, the radius of star gets smaller until gravitational force can take the control of particles going out of star which in turn creates a black hole or a neutron star if the initial mass of star is below the Chandrasekhar limit (CL). Otherwise the star becomes a white dwarf with a final mass again below CL. But I'm asking, assuming that the mass of star is less than CL, why the storage of energy in a very huge amount produced through nuclear fusion does not lead to the collapse of star, say, like a car blast, and doesn't this mean that energy is not conserved inside a star which loses mass while the particles velocity remains bounded? Is this even meaningful?

AB

 

[Ich]

ok, either I completely misunderstand what you're saying, or you're starting from a number of major misconceptions about the topic of this thread (degeneracy pressure) and energy conservation. Maybe somebody else, who understands better what you're saying, wants to answer.

 

[Naty1]

bcrowell:

Once the speeds become relativistic, you can no longer decrease this time between impacts by adding energy.

Can we also attribute this to the Pauli exclusion principle ?? In your write up you discuss Pauli at norelativistic speeds...and I assume it continues at relativistic speeds.

Thats a real nice way to think about this. Also, I am guessing that all this is where mostly quantum mechanics applies and einstein's GR gets real fuzzy...His pressure,momentum,energy relationships would likely be different at this scale,right??
Do

Also, I read a bit further and came across electron capture! Holy cow it's been along, long time since I studied a bit of that in the dark ages and had forgotten it entirely...
And I see you have some tensor discussion prior to 4.4.3..I'll check that out because at other locations when attempting to learn tensors on my own, my attention span proved shorter than the concepts!
Thanks ...

 

[Naty1]

Post #7 makes virtually no sense to me..so if it does to others, an interpretation would be helpful...

edit: I just saw Ichs comments..I agree...post #7 seems riddled with misconceptions.

separately,
Does relativistic quantum mechanics (or any other kind) tell us anything heat production at these ultra compact densities?

Seems like even absent thermonuclear reactions there would be substantial heat produced yet I have not seen it mentioned. So I'm guessing that as the electrons begin to act as a degenerate gas and their collapse slows or halts, whatever heat is generated just dissipates. If the star is small enough it just sits cold and dark; otherwise it collapses into a black hole...and as far as I know they are so cold its theorized all of them are still absorbing radiation from "our" space...they are colder the the CMBR.

 

[Altabeh]

Post #7 makes virtually no sense to me..so if it does to others, an interpretation would be helpful...

edit: I just saw Ichs comments..I agree...post #7 seems riddled with misconceptions.

Could someone specify these misconceptions so I can be able to correct them? I'm not basically an astrophysicist, and I just asked for more information upon the main issue and the one brought up by bcrowell through this sentence:

Once the speeds become relativistic, you can no longer decrease this time between impacts by adding energy

This comes right after If this time is short, then there are more impacts meaning that if time doesn't get shorter, then there are less impacts happening in the system which sounds logical but is in contrast to the above quote. Because adding more energy into system would lead to more impacts, I think, if the system is not going to collapse otherwise the energy would not be conserved!

AB

 

[Ich]

Could someone specify these misconceptions so I can be able to correct them?

Whats hardest not to interpret as a misconception is this:

But as the heat carries energy, so the energy of system increases, giving rise to the increase of the number of particles crossing through the surface of star.

The energy of the system doesn't increase. Nuclear reactions change mass into heat, the sum is the same - except that radiation and solar wind carry some energy away.
Then, pressure is not the number of particles passing through a star's surface. You pick any surface within the star, count how much momentum flows from one side to the other and back, and that's the definition of pressure at said surface.
With a given particle density, an increase in momentum p changes two things:
1) the momentum per particle increases ~p (obviously)
2) the velocity increases ~p for a slow particle, but stays ~=c for very fast particles

So when all particles travel at nearly the speed of light, you can't make them faster. Thus, the number of particles crossing through the surface does no longer increase with momentum.

Then, we're talking about a degenerate gas. That's where a few million K count as "cold".
Heat is totally neglected, the particle momentum is dictated only by the Heisenberg uncertainty principle, as ~number density n³.
So you have pressure = n*(momentum per particle)*(particle speed), which changes as
dP~ dn^(1+1/3+1/3) for slow particles, and
dP~dn^(1+1/3+0) for relativistic particles.

 

[Altabeh]

Whats hardest not to interpret as a misconception is this:

The energy of the system doesn't increase. Nuclear reactions change mass into heat, the sum is the same - except that radiation and solar wind carry some energy away.
Then, pressure is not the number of particles passing through a star's surface. You pick any surface within the star, count how much momentum flows from one side to the other and back, and that's the definition of pressure at said surface.
With a given particle density, an increase in momentum p changes two things:
1) the momentum per particle increases ~p (obviously)
2) the velocity increases ~p for a slow particle, but stays ~=c for very fast particles

So when all particles travel at nearly the speed of light, you can't make them faster. Thus, the number of particles crossing through the surface does no longer increase with momentum.

Then, we're talking about a degenerate gas. That's where a few million K count as "cold".
Heat is totally neglected, the particle momentum is dictated only by the Heisenberg uncertainty principle, as ~number density n³.
So you have pressure = n*(momentum per particle)*(particle speed), which changes as
dP~ dn^(1+1/3+1/3) for slow particles, and
dP~dn^(1+1/3+0) for relativistic particles.

Now I got it! You know, as this first was brought up in General Relativity forum, as someone looking strictly from the standpoint of conservation law of energy, I thought by "adding" bcrowell meant some sort of external energy (for instance, infalling particles/bodies hitting the star) would be injected into the system and this energy happens to expose itself as heat so if this does not affect system then...!

Anyways, thanks for clarification and one more question: How does one really get dP~ dn^(1+1/3+1/3) or dP~dn^(1+1/3+0)? Is this based upon experiments or it just has some mathematical reasoning behind it?

AB

 

[Ich]

How does one really get dP~ dn^(1+1/3+1/3) or dP~dn^(1+1/3+0)?

You assume that the particles' momentum is defined by the uncertainty principle.
With V=Volume per particle = 1/n, n=1/V=number density=particles per Volume, p=momentum per particle you get
hbar/2 = const. = x*p = V^(1/3)*p ~ n^(-1/3)*p, so
p~n^(1/3),
n~p^3
Then, as explained before, pressure
P ~ n*(momentum per particle)*(particle velocity) =
= p^3*p*p = n^(5/3) if velocity ~ p
= p^3*p*1 = n^(4/3) if velocity = const. = c

 

[twofish-quant]

Once the speeds become relativistic, you can no longer decrease this time between impacts by adding energy.

Just to amplify this a bit, the reason you can't decrease the time between impacts is that no matter how much energy you add to the system, the particles are not going to move back and forth at faster than the speed of light.

This has a very important implication. You can change the particles that are rattling back and forth, but no matter what the particles are, they can't move back and forth at faster than the speed of light, and so the time between impacts is always going to be bounded. Which means that no matter what the particles are, eventually if you pile on enough matter, it's going to collapse.

 

[twofish-quant]

But to my own knowledge, the more energy you add to any material system, the faster its particles oscillate so the more impacts will happen.

There is a limit to how fast particles can oscillate. They can't move faster than light, which is where relativity comes in.

 

[Ich]

1000 posts in 125 days and science advisor. Congratulations.

 

[Altabeh]

Which means that no matter what the particles are, eventually if you pile on enough matter, it's going to collapse.

This was my point, too!

And thank you Ich for your explanation. But one bothering thing is that why would one make use of the Uncertainty inequality as an equality? I think such usages of UP are extensively approximate and won't give much reliable estimations of real numbers!

AB

 

[Naty1]

Originally Posted by Altabeh

But to my own knowledge, the more energy you add to any material system, the faster its particles oscillate so the more impacts will happen

.

Did you read the reference given in post #3??...that gives some understanding of why this effect decreases as any material is compressed to such high densities that relativistic electrons can no longer follow their macroscopic behavior...

But this quantum behavior is by no means at all obvious...

 

[Altabeh]

1000 posts in 125 days and science advisor. Congratulations.

I also send my congrats to twofish-quant for his useful notes! I don't know what field he is active in, but hope for him to stay with us here for as long as he can!

AB

 

[Matterwave]

This was my point, too!

And thank you Ich for your explanation. But one bothering thing is that why would one make use of the Uncertainty inequality as an equality? I think such usages of UP are extensively approximate and won't give much reliable estimations of real numbers!

AB

Degeneracy only really comes into play when the HUP is close to equality, so it's not a bad approximation to make. If the HUP is not close to equality, then the particles need not be degenerate, they would have more states to move around in.

 

[Altabeh]

Degeneracy only really comes into play when the HUP is close to equality, so it's not a bad approximation to make. If the HUP is not close to equality, then the particles need not be degenerate, they would have more states to move around in.

So it must mean that either the mumentum of particles of matter (gas) or their position is highly uncertain because of the extremely high density, right?

AB

 

[twofish-quant]

So it must mean that either the mumentum of particles of matter (gas) or their position is highly uncertain because of the extremely high density, right?

Not directly. What happens is that degeneracy pressure arises when because the Pauli exclusion principle says that you can't have two electrons in the same quantum state. When the density is low, this restriction doesn't matter. It matters only when all the electrons are packed together so that there is only one electron in each quantum state.

One way of thinking about it is to imagine each quantum state as a box. When you have a few electrons, they get scattered across a lot of different boxes, so that you don't notice that the boxes exist. When you have a lot of electrons, the boxes are all filled up so the size of the box matters. The Heisenberg uncertainty principle tells you how big the boxes are, and that's how it enters into the picture.

 

[Naty1]

So it must mean that either the mumentum of particles of matter (gas) or their position is highly uncertain because of the extremely high density, right?

I think that's a reasonable way to think about confinement.

But I like twofish's reply. A similar way to describe the phenomena is that while you might expect the location to become more certain via classical reasoning as matter gets compressed, in fact you are thwarted by ever higher energy levels resulting from quantum confinement. So maybe one could say more accurately that at high densities momentum and position remain uncertain.

 

[Naty1]

Here's a good discussion, don't know why I did not look here first:


Wikipedia sums it up nicely :

Electron degeneracy pressure is a consequence of the Pauli exclusion principle, which states that two fermions cannot occupy the same quantum state at the same time.

here

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

and has some insights we did not post...like

This degeneracy pressure is omnipresent and is in addition to the normal gas pressure P = nkT / V

Also a related discussion is here:
http://en.wikipedia.org/wiki/Degenerate_matter

 

[qraal]

Here's a good discussion, don't know why I did not look here first:


Wikipedia sums it up nicely :
here

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

and has some insights we did not post...like


Also a related discussion is here:
http://en.wikipedia.org/wiki/Degenerate_matter

All Hail Wikipedia!

Frequently gives me the best summary and the best leads on mathematical tough-nuts I'm trying to crack... though its piece on Dawson's Integral is a bit too summarized and the leads aren't the best for working out a numerical computation. Might have to edit the page when I find some good online stuff the hard way... ;-)