Are there widespread misconceptions about degeneracy pressure?

[Ken G]

Two statements that are often made about degeneracy pressure are:
1) It is a new or special kind of pressure that requires quantum mechanics, in contrast with ideal gas pressure, which in effect involves somewhat mysterious forces that emerge from the Pauli exclusion principle, and
2) it behaves in such a way that degenerate gases do not expand like ideal gases when heat is added to them, which allows the heat to build up and fusion to run away (which causes helium flashes and type Ia supernovae).

For example, these claims can be found in many textbooks, and in Wiki:
http://en.wikipedia.org/wiki/Electro...eracy_pressure
"The Pauli exclusion principle disallows two half integer spin particles (fermions) from simultaneously occupying the same quantum state. The resulting emergent repulsive force is manifested as a pressure against compression of matter into smaller volumes of space."
So that certainly sounds like a "T" for (1). Then we have:
http://en.wikipedia.org/wiki/Type_Ia_supernova
"A main sequence star supported by thermal pressure would expand and cool in order to counterbalance an increase in thermal energy. However, degeneracy pressure is independent of temperature; the white dwarf is unable to regulate the burning process in the manner of normal stars, and is vulnerable to a runaway fusion reaction."
If the point being made here seems unclear, it is often explained further in the quite similar conditions that appear in a helium flash:
http://en.wikipedia.org/wiki/Helium_flash
"A helium flash occurs in these situations because the helium is degenerate, meaning it is supported against gravity by quantum mechanical pressure rather than thermal pressure. Thus an increase in the temperature in the material undergoing fusion does not act to expand the material and by doing so cool it, and there is no regulation of the rate of fusion. "

Certainly there are always idealizations and generalizations needed to simplify complex physics, but do the above two statements about degeneracy pressure really encapsulate the essence of the phenomena encountered, or are they pretty much false myths that are propagated simply because they are not subjected to critical scrutiny? What do people think? The poll possibilities are TT, which is both statements are mostly true, or TF, so statement (1) is mostly true but statement (2) is basically a myth, or FT if the opposite, or FF if both statements are mostly myths that do more to foster misconceptions than bring insights.

 

[Ken G]

No opinions on the matter? Are the people on this forum not very familiar with degeneracy pressure? You could read the cited Wiki article, and see if you agree with the two statements in the OP. The physics is elementary quantum mechanics, but there are interesting subtleties when it comes to the interpretation, hence the need to gauge the value of those two claims.

 

[alialice]

The two statements are both true.
The second one, related to helium flash, can be explained with the concept of thermal stability. If a degenerate system goes under compression, it cools down (it becomes "more degenerate" because density increases). Thermal stability is achieved only by classical system, for which:

sign(d T) = sign (d rho)

d =: "variation of"
T=central temperature
rho=central density

 

[Ken G]

So you would then vote "TT" in the poll. You are certainly not alone, many textbooks make similar claims. But if we look at them more carefully, are they actually correct? How many forum members see those statements as true, or do some find falsehood there? If you analyze more carefully, I wager you will see the situation, at best, is not so cut and dried, and at worst, is hopelessly confused by those two claims. (Let me interject that I agree with your statement that degenerate gases are thermally unstable and ideal gases are thermally stable, if a global force balance is maintained the whole time, but the issue is--- why?)

 

[Ken G]

Perhaps degeneracy pressure is not of great interest in this forum, it is a primarily astrophysical topic. All the same, both statements made about it in the OP are essentially completely false, even though they are propagated widely. Degeneracy is an entirely thermodynamic effect, it has no mechanical consequences that distinguish it in any way from an ideal gas. That is, it is a constraint on the ratio of temperature to energy per particle, but it has no effect on pressure whatsoever in any situation where the energy per particle is already specified. As such, it is a perfectly garden variety pressure, if the processes that set the energy are already being tracked. Where degeneracy is important is in the thermodynamics, that is, when we want to track the heat transport via knowledge of the temperature. That will of course affect the kinetic energy and the pressure of the gas, but that's the only place where it connects with pressure.

What's more, it is completely false that degenerate gases do not expand when heat is added to them, they expand exactly the same as ideal gases do. This is an elementary result, derivable from the very same equations you will find in the textbooks that say degenerate gas doesn't expand when it is heated. To see this, what is required is more precise thermodynamic usage, where "heat" and "temperature" are not treated as interchangeable concepts. The thermal instability of a degenerate gas has nothing at all to do with the gas not expanding, and indeed it is patently false that the gas does not expand. I would be happy to expound on these points if people are curious, the treatment is all undergraduate level physics.

 

[alialice]

What's more, it is completely false that degenerate gases do not expand when heat is added to them, they expand exactly the same as ideal gases do. This is an elementary result, derivable from the very same equations you will find in the textbooks that say degenerate gas doesn't expand when it is heated. To see this, what is required is more precise thermodynamic usage, where "heat" and "temperature" are not treated as interchangeable concepts. The thermal instability of a degenerate gas has nothing at all to do with the gas not expanding, and indeed it is patently false that the gas does not expand. I would be happy to expound on these points if people are curious, the treatment is all undergraduate level physics.

Why do you say that? So how do you explain helium flash? After the nuclear explosion the gas is no more degenerate and it expand regularly.

 

[alialice]

That is, it is a constraint on the ratio of temperature to energy per particle, but it has no effect on pressure whatsoever in any situation where the energy per particle is already specified. As such, it is a perfectly garden variety pressure, if the processes that set the energy are already being tracked. Where degeneracy is important is in the thermodynamics, that is, when we want to track the heat transport via knowledge of the temperature. That will of course affect the kinetic energy and the pressure of the gas, but that's the only place where it connects with pressure.

I surely refer to a thermodynamic interpretation in what I'm goin to write. I don't agree on the point where you say that degeneracy has no effect on pressure, or that it has only in connection with heat transport. White dwarfs and neutron stars are maintained by degeneracy pressure because they the correct density and temperature relation to obtain this condition. And the degenerate component in most case doesn't correspond with the responsible of heat transfer.

 

[cristo]

NB. Polls are not permitted in the science forums, since they do not give any added benefit to the thread.

 

[Ken G]

Why do you say that? So how do you explain helium flash? After the nuclear explosion the gas is no more degenerate and it expand regularly.

It expands the same when heat is added, it makes no difference if it is ideal or degenerate. The helium flash has nothing to do with presence or absence of expansion, and the way to see that is to ask, would there be a helium flash if the helium was as degenerate as the electrons? The answer is no-- yet nothing you will typically find in the erroneous explanations for the helium flash can account for this uncontroversial fact.

 

[Ken G]

I surely refer to a thermodynamic interpretation in what I'm goin to write. I don't agree on the point where you say that degeneracy has no effect on pressure, or that it has only in connection with heat transport. White dwarfs and neutron stars are maintained by degeneracy pressure because they the correct density and temperature relation to obtain this condition. And the degenerate component in most case doesn't correspond with the responsible of heat transfer.

I would argue that it is very misleading to claim that degeneracy causes the pressure in white dwarfs. Degeneracy is not the reason a white dwarf is small, that is simply due to its history of losing heat. Degeneracy is also not the reason that the particles have lots of kinetic energy, that is due to the virial theorem, indeed it is a trivial example of the virial theorem. The only reason that degeneracy matters in a white dwarf is that it shuts off the thermodynamic heat transfer from the degenerate gas to its environment, which prevents further contraction. That is not a cause of pressure.

 

[zonde]

There certainly are widespread misconceptions about degeneracy pressure.

Two statements that are often made about degeneracy pressure are:
1) It is a new or special kind of pressure that requires quantum mechanics, in contrast with ideal gas pressure, which in effect involves somewhat mysterious forces that emerge from the Pauli exclusion principle, and
2) it behaves in such a way that degenerate gases do not expand like ideal gases when heat is added to them, which allows the heat to build up and fusion to run away (which causes helium flashes and type Ia supernovae).

First statement is certainly false but it is a bit more complicated with second. The idea that you can add a lot of energy without getting much pressure in response is because particles become relativistic i.e. energy is going up but speed of particles is bound to be no more than c. I believe the argument had some more steps before you arrived at contraction. Something about increasing gravity.

 

[Ken G]

You are right that things change as the particles go relativistic, but note that the helium flash is often the place where these kinds of arguments about degeneracy pressure appear, and that stays pretty nonrelativistic, though it's only an approximation so one needs to choose how accurate one wants to be.

 

[zonde]

Wikipedia page about helium flash does not seem very clear.
It speaks about degeneracy pressure when talking about core helium flash. But it says that "The [core] helium flash is not directly observable on the surface by electromagnetic radiation." So it's hypothetical phenomena.
In case of observable helium flashes wikipedia page does not give much of the explanation.

 

[Ken G]

Yes, the "flash" is too deep in the star to observe, it is just a theoretical expectation. Still, the transition from a red giant to a "horizontal branch" star is observed, and that is supposed to be the change that the helium flash brings about, but still it is just a theoretical step whose details are probably not well known. I'm referring more to our pedagogical understanding of what should be happening, more so than any details that can be observed!

 

[Drakkith]

I don't quite understand your questions, Ken. Per wiki: http://en.wikipedia.org/wiki/Helium_flash

The explosive nature of the helium flash arises from its taking place in degenerate matter. Once the temperature reaches 100 million–200 million kelvins and helium fusion begins using the triple-alpha process, the temperature rapidly increases, further raising the helium fusion rate and, because degenerate matter is a good conductor of heat, widening the reaction region.
However, since degeneracy pressure (which is purely a function of density) is dominating thermal pressure (proportional to the product of density and temperature), the total pressure is only weakly dependent on temperature. Thus, the dramatic increase in temperature only causes a slight increase in pressure, so there is no stabilizing cooling expansion of the core.
This runaway reaction quickly climbs to about 100 billion times the star's normal energy production (for a few seconds) until the temperature increases to the point that thermal pressure again becomes dominant, eliminating the degeneracy. The core can then expand and cool down and a stable burning of helium will continue.[1]

This seems pretty clear that the addition of thermal energy only adds a small amount of pressure at first, so the temperature increases many many times what it was, leading to an "explosive" burning of helium as the temperature skyrockets, lasting until the gas can expand and cool off once more.

Are you saying this is wrong?

 

[Ken G]

Yes, I'm saying that the text in bold is pretty close to completely wrong. The only reason it isn't 100% wrong is that it tends to focus its attention on temperature, and indeed the temperature response of a degenerate gas is a bit bizarre. However, it fails to describe what is happening with energy, and of course following the energy is always a crucially important thing to do in physics. To expose the flaws in the bold text, simply ask this question: does the helium flash happen if helium is just as degenerate as the electrons? It is easy to see that the answer is no, yet what part of that Wiki explanation would suggest even in the least bit that no helium flash occurs if helium is also degenerate?

What is actually happening in a helium flash is infinitely more interesting than that description. The first thing to get is that the pressure in any nonrelativistic gas, be it ideal or degenerate, is 2/3 the kinetic energy density. This is an elementary result, I'm sure you can derive it in ten seconds. So if you follow the energy, it becomes much more clear that adding heat to a degenerate gas creates exactly the same expansion as for an ideal gas. There is no need to wait for the gas to become ideal, expansion comes with energy deposition by fusion, period. Now, it is true that the temperature spikes much more rapidly in a degenerate gas, but here's the interesting part-- the temperature rises even though the internal energy per particle drops (due to expansion work). The latter is elementary, the T rise is what is subtle and relies on degeneracy.

But now here's the kicker-- if the helium is also degenerate, then the T rise would not correspond to raising the energy of the helium, so would not cause more fusion. The gas would be thermally stable, and for exactly the same reason that hydrogen fusion in the Sun is stable-- adding heat would cause expansion, which would do work, remove internal energy, and shut off the helium burning. Where is that accounted for in the essentially incorrect bolded text above?

(ETA: the point is, the bottom line is that the helium flash has to do with the strange temperature behavior of degenerate gas, which the Wiki quote does allude to, but it has nothing whatsoever to do with anything going on with pressure. What is happening with pressure is completely mundane, and is the same as for an ideal gas. The issue is all thermodynamic, pressure and expansion are red herrings.)

 

[Drakkith]

I'm sorry, Ken, I don't know enough to really understand you.

To expose the flaws in the bold text, simply ask this question: does the helium flash happen if helium is just as degenerate as the electrons?

I have no idea. Are you asking if a helium flash happens if the ions are degenerate like the electrons are? If so, why does that even matter in this situation?

The first thing to get is that the pressure in any nonrelativistic gas, be it ideal or degenerate, is 2/3 the kinetic energy density.

Is degenerate gas in the core of a star a nonrelativistic gas?

So if you follow the energy, it becomes much more clear that adding heat to a degenerate gas creates exactly the same expansion as for an ideal gas.

I cannot follow the energy, and what you're saying goes against everything I've read so far. Can you show some math or something?

 

[Ken G]

Are you asking if a helium flash happens if the ions are degenerate like the electrons are? If so, why does that even matter in this situation?

It's not so much that the words in the Wiki quote are literally incorrect, it is that they do not convey the real reason there is a helium flash. Pressure plays no role at all, neither does expansion. The key issue is that adding heat causes the temperature of a degenerate gas to rise, because it softens the degeneracy. But the energy per particle of the degenerate gas falls, because of expansion-- the expansion that is not supposed to be happening. However, that doesn't matter, because if the temperature rises, the energy of the ideal helium nuclei goes up. It's all about breaking degeneracy, expansion doesn't matter a whit.

Is degenerate gas in the core of a star a nonrelativistic gas?

To a decent approximation, yes, at least for the helium flash.

I cannot follow the energy, and what you're saying goes against everything I've read so far. Can you show some math or something?

Let's agree that pressure is 2/3 the kinetic energy density, ideal or degenerate. Are we good there?

 

[Drakkith]

It's not so much that the words in the Wiki quote are literally incorrect, it is that they do not convey the real reason there is a helium flash. Pressure plays no role at all, neither does expansion. The key issue is that adding heat causes the temperature of a degenerate gas to rise, because it softens the degeneracy. But the energy per particle of the degenerate gas falls, because of expansion-- the expansion that is not supposed to be happening. However, that doesn't matter, because if the temperature rises, the energy of the ideal helium nuclei goes up. It's all about breaking degeneracy, expansion doesn't matter a whit.

What I'm getting from this:

Electron energy per particle falls because it is expanding and becoming non-degenerate.
Nuclei energy increases because the temperature is increasing.

Is that correct?

To a decent approximation, yes, at least for the helium flash.Let's agree that pressure is 2/3 the kinetic energy density, ideal or degenerate. Are we good there?

Uhh, sure?

 

[Ken G]

What I'm getting from this:

Electron energy per particle falls because it is expanding and becoming non-degenerate.
Nuclei energy increases because the temperature is increasing.

Is that correct?

Yes, exactly.

Uhh, sure?

The point there is that if degeneracy doesn't affect the pressure once we specify the energy density, then it also doesn't affect the pressure once we say how much heat is being added by fusion. So pressure is a complete red herring, it has nothing to do with the interesting things that degeneracy is doing in the helium flash. The clear misconception fostered by the Wiki quote, and a million other places, is that degenerate gas doesn't expand when heat is added, so the heat piles up. That's what is wrong (although that particular Wiki quote doesn't actually say that, I'll give it credit for that). The heat does not pile up, but the temperature does rise. That's what we need to understand about degeneracy-- how it let's the temperature rise even as the average energy per electron is dropping. Nothing in the Wiki quote conjures that crucial state of affairs, so it really misses the boat on what is subtle and interesting about degeneracy. It just has nothing to do with pressure, and neither does the helium flash.

 

[Drakkith]

That's what is wrong (although that particular Wiki quote doesn't actually say that, I'll give it credit for that). The heat does not pile up, but the temperature does rise. That's what we need to understand about degeneracy-- how it let's the temperature rise even as the average energy per electron is dropping. Nothing in the Wiki quote conjures that crucial state of affairs, so it really misses the boat on what is subtle and interesting about degeneracy. It just has nothing to do with pressure, and neither does the helium flash.

How does the temperature rise if the heat isn't "piling up"? (Not really even sure what that means)

Also, perhaps it is confusing because the electrons are degenerate while the ions are not. How does temperature work in this case where there are two clearly different states; the degenerate electrons and the non-degenerate ions.

 

[Ken G]

How does the temperature rise if the heat isn't "piling up"? (Not really even sure what that means)

It means the internal energy of the gas is dropping throughout the helium flash, the way it is normally modeled. The reason it is dropping is exactly the process that is often said is not happening-- expansion work. The temperature rises because that's what happens when degenerate gas is heated, expands, and has its internal energy drop. This is what we need to understand about degenerate gas, its rather unusual thermodynamics. Focusing on pressure is exactly the wrong place to look, nothing is happening with pressure that matters at all.

Also, perhaps it is confusing because the electrons are degenerate while the ions are not. How does temperature work in this case where there are two clearly different states; the degenerate electrons and the non-degenerate ions.

Each has its own way of arriving at a temperature, and this is the crucial issue in the helium flash. It's thermodynamics, not mechanics.

 

[Drakkith]

Tell me if the following is correct:

The pressure in a degenerate gas depends only on the speed of the degenerate particles NOT the temperature of the gas. But to change the speed of degenerate particles requires A LOT of energy because they are locked into place against each other. Adding heat only causes the non-degenerate particles to move faster, but the degenerate ones supplying the pressure are unaffected.


From: http://www.astronomynotes.com/evolutn/s10.htm (Step 2)

I assume they are mostly unaffected, not completely unaffected?

 

[Ken G]

Tell me if the following is correct:

The pressure in a degenerate gas depends only on the speed of the degenerate particles NOT the temperature of the gas. But to change the speed of degenerate particles requires A LOT of energy because they are locked into place against each other. Adding heat only causes the non-degenerate particles to move faster, but the degenerate ones supplying the pressure are unaffected.

I would say that quote misses the mark even worse than the Wiki quote. It is just loaded with the potential for fostering misconceptions, I really don't see much of value in it to be quite honest. This is why I started with the point that the pressure is 2/3 the kinetic energy density, whether the gas is ideal or degenerate. Once you recognize that, you really have to wonder why this quote is trying to suggest that somehow adding heat to a degenerate gas doesn't raise the pressure like it would an ideal gas. It's all just a question of how much heat you are adding compared to how much internal energy is already there, none of that has anything directly to do with degeneracy, nor tells us anything interesting about degeneracy. Yes a white dwarf has a lot of internal energy, the virial theorem tells you that. It would have the same internal energy if it were an ideal gas of bosons of the same mass and radius. This is just missing the issue of what degeneracy does-- it makes the temperature very low compared to an ideal gas of bosons, all else being equal. There just are no pressure implications, that's a complete red herring.

 

[Drakkith]

I think I'm just more confused now than before I got into the thread.

 

[Ken G]

That's because you have been told a lot of hooey about degeneracy pressure that you first have to unlearn!

Let me give you a concrete example. We have a nonrelativistic gas with pressure P and density n. We then know its average energy per particle, that is U = 3/2 P/n. We say the gas is self-gravitating and obeys the virial theorem, which means that changes in its internal energy will always obey dU/U = -dR/R = 1/3 dn/n. This is saying the self-gravitating gas is staying in a homologous force balance as n changes, which is a standard assumption used to analyze all the things we have been talking about.

Now say we add heat per particle dQ. The first law of thermodynamics says dU = dQ + P dn/n² = dQ + 2/3 U dn/n = dQ + 2 dU. Hence we derive dU = -dQ, the standard outcome of the virial theorem. It says that if you add heat to a self-gravitating non-relativistic gas, you reduce the internal energy by the amount you think you should be increasing it by. This reduction is due to expansion work.

Now here comes the kicker: where did I need to say if this gas was ideal or degenerate? I didn't, I never had to say which it is-- because the whole issue of degenerate vs. ideal behavior has nothing to do with how the gas responds to having heat added, unless I need to talk about the temperature. Since I didn't need to talk about temperature to derive dU = -dQ, it has nothing to do with degeneracy. Neither does dn/n = 3 dU/U = -3 dQ/U. So this is what I mean, degeneracy has nothing to do with the way pressure responds to heat being added, and nothing to do with how the gas expands when heat is added. Degeneracy is about a rather bizarre type of temperature behavior, and a lot of what is written about it completely obfuscates this important point.

The Wiki quote, and those course notes, are classic examples of the mire you get into when you mix the things that don't care about degeneracy with the things that do. Degeneracy has no pressure implications in any situation where you have already specified the energy density and particle density, it's just that some combinations of those parameters might make it impossible to find a physically possible temperature if the gas is fermionic, so those energy densities are impossible for degenerate gases of given density. But if you have a possible energy density and particle density, and you don't care what the temperature is doing (as for solving the expansion and the internal energy change, as above), then degeneracy plays no role whatsoever. I could have just said: degeneracy plays no role at all in the virial theorem.

 

[zonde]

the pressure is 2/3 the kinetic energy density, whether the gas is ideal or degenerate.

It's wrong.
Look, kinetic theory explains pressure as result of collisions between particles or collisions of particles with container. But collision change quantum states of colliding particles. So outcome of collision is restricted by available quantum states. And if there are no accessible quantum states there are no collisions and no pressure. Degenerate particles just move on their way no matter how much kinetic energy they have.

 

[Ken G]

But you are simply demonstrating that I am correct in the title of this thread. I don't blame you for holding that false impression, there are so many places that say degeneracy pressure is some strange kind of quantum mechanical force that appears when you have a Pauli exclusion principle. But it's all baloney-- degeneracy pressure is gas pressure, and it is very definitely 2/3 the kinetic energy density in any nonrelativistic gas. The same sites that make the spurious claims about it will usually invoke this fact at some point.

Actually, it's kind of an interesting question as to why you just said is wrong. One might imagine that collisions would be changed by the availability of states, and indeed degenerate gas is highly conducting of heat and can even act as a superfluid (though I don't know many details about that). But collisions between particles has nothing to do with gas pressure, collisionless gases have the same pressure as collisional ones do. It's all about kinetic energy density, that's an elementary result of the definition of pressure.

 

[Drakkith]

Are you saying degeneracy pressure is gas pressure because it is the result of the kinetic energy of the electrons, like normal gas pressure is the result of the kinetic energy of the gas particles?

 

[Ken G]

Yes.

 

[Drakkith]

Yes.

Oh good, I was hoping I'd put everything together correctly.

Any chance you can answer the following?If we have a degenerate stellar core at around 100 million k, and we increase the temperature to 200 million k, what exactly happens?

• How much might the core expand?

• What would the thermal pressure of the ions be compared to the pressure of the degenerate electrons?

• How would the increase in energy be distributed between the ions and electrons?

 

[zonde]

But you are simply demonstrating that I am correct in the title of this thread.

If what I say is wrong then they are my more or less private misconceptions. Certainly not widespread.

I don't blame you for holding that false impression, there are so many places that say degeneracy pressure is some strange kind of quantum mechanical force that appears when you have a Pauli exclusion principle. But it's all baloney-- degeneracy pressure is gas pressure, and it is very definitely 2/3 the kinetic energy density in any nonrelativistic gas. The same sites that make the spurious claims about it will usually invoke this fact at some point.

Actually, it's kind of an interesting question as to why you just said is wrong. One might imagine that collisions would be changed by the availability of states, and indeed degenerate gas is highly conducting of heat and can even act as a superfluid (though I don't know many details about that). But collisions between particles has nothing to do with gas pressure, collisionless gases have the same pressure as collisional ones do. It's all about kinetic energy density, that's an elementary result of the definition of pressure.

Definition of pressure is force per unit area applied in a direction perpendicular to the surface of an object.
Without collisions how is force applied to anything? How is energy transferred from the gas to the object that is measuring force?

 

[Ken G]

If we have a degenerate stellar core at around 100 million k, and we increase the temperature to 200 million k, what exactly happens?

Temperature is very complicated with degenerate gas, that's the issue. It is much easier to know what happens to the internal energy if you add heat (dU = -dQ). To solve for the temperature requires solving for the degeneracy parameter, it's a bit tedious but is in standard references on the degeneracy parameter. It acts a lot like a thermodynamic chemical potential. But I'll give you basic non-quantitative answers:

• How much might the core expand?

This depends on just how degenerate the gas is originally. Assuming it's highly degenerate originally, not much expansion will occur-- you won't have to add much heat to get the temperature to rise. Then to double the temperature, all you have to do is get the degeneracy parameter to drop by a factor of 2, but if it is already very high, it will still be highly degenerate. So it won't change the gas much. Note this is what I have been saying-- degeneracy is a temperature effect, so changing the temperature is about changing the degeneracy. Adding heat will change the degeneracy, which will change the temperature. Expansion is irrelevant, and it is false to say the gas doesn't expand when you add heat. If you track the heat added, the expansion is identical to an ideal gas. If you track the change in temperature, the degenerate situation is completely different, because degeneracy is all about temperature.

• What would the thermal pressure of the ions be compared to the pressure of the degenerate electrons?

The ion pressure will double if this happens with little expansion, but will still be way less than the electron pressure, since we are assuming the degeneracy is staying high.

• How would the increase in energy be distributed between the ions and electrons?

This is the crux of the helium flash-- if you add a unit of heat, the total internal energy goes down by 1 unit (as required by the virial theorem), but the electron energy goes down by 2 units and the ion energy goes up by 1 unit. Internal energy passes from the electrons to the ions, this is the essential cause of the helium flash that you will basically not find in any textbook because they have all bought off on the myth that the helium flash has something to do with a lack of expansion work being done, which it does not.

 

[Ken G]

If what I say is wrong then they are my more or less private misconceptions. Certainly not widespread.

Well, your take on the issue is not necessarily in the textbooks, but it is already more deeply related to the actual physics of degeneracy than what you normally find there. In all honesty, I have as yet not spoken with a single person who really understands degeneracy, and neither did I just a few short months ago. The misconceptions are very widespread.

Definition of pressure is force per unit area applied in a direction perpendicular to the surface of an object.
Without collisions how is force applied to anything? How is energy transferred from the gas to the object that is measuring force?

Pressure is a diagonal stress-energy tensor. If you look up the definition of the stress-energy tensor, you will see no reference to any collisions anywhere. That's another widespread myth about pressure. The main thing to get is that pressure gradients produce forces on fluids, which simply means, gradients in momentum fluxes generate momentum deposition when you average over the fluid. The momentum deposition has nothing to do with collisions, it is just how momentum flux gradients work, they yield momentum piling up in a volume. The thing you need collisions for is to keep the fluid behaving nicely, like with locally isotropic distribution functions and so forth (so the stress-energy tensor stays diagonal and pressure takes on its simple meaning). You don't even need collisions off a boundary, the force produced by pressure is perfectly capable of acting on the fluid itself and not anything else. Indeed, I think you raise an interesting point, that something quite strange must occur when a degenerate gas encounters a wall. The problem is that you can no longer treat them as being in momentum eigenstates if you have a wall, so the energy eigenstates are not momentum eigenstates any more and life gets a bit complicated, but I presume they induce the normal pressure that an ideal gas would at the same energy density.

 

[Drakkith]

What's the difference between increasing the temperature and adding heat?