Undergrad Why is the Pauli Exclusion Principle not a force?

[my2cts]

Well, you have seen it here.

No I have not. I have seen an explanation of degeneracy pressure in terms of momentum, but that is not the full answer.
That was a good point.
The question that remains is why the momentum is so high. There is no answer to that question here.

[Moderator's note: off topic comments deleted.]

 

[PeterDonis]

The question that remains is why the momentum is so high. There is no answer to that question here.

The answer is: Because the fermions are confined to a finite region of space by some potential. But the only way to know what specific potential, and therefore what specific form the momentum will take, is to look at the details of the specific system (e.g., a hydrogen atom will be very different from a metal).

 

[Ken G]

No I have not. I have seen an explanation of degeneracy pressure in terms of momentum, but that is not the full answer.

Yes, it really is. Some real forces give momentum to the particles, and that momentum flux density is what we call pressure. All the PEP does is constrain how that momentum can be given to the particles, it is not a source of any momentum so that answers the OP question. The OP also asked about exchange energy, but again, that is not energy that the PEP provides, it is provided by the actual forces, but the PEP helps constrain how much energy those forces provide by controlling the allowed states.

The question that remains is why the momentum is so high. There is no answer to that question here.

Sure there is, the answer is that the momentum is not necessarily high. It is only high if the forces that are present in some specific situation deposit a huge amount of momentum, as always happens where there is a huge degree of compression.

 

[Jilang]

[QUOTE="my2cts, post: 5647534, member: 488871
The question that remains is why the momentum is so high. There is no answer to that question here.[/QUOTE]
That was the point of my answer in post #84. With fermions when the lower energy levels are filled only the higher ones are available. If the volume increases the number of possible ground states decreases. So to reduce the volume further some will need to be promoted to higher energy levels (wth higher momentum). To do this requires energy. When you talk about what pushes back, it is the same as what resists acceleration...inertia.

 

[Ken G]

I think what a lot of people don't understand is what would happen if you waved a magic wand over a white dwarf and made all the electrons distinguishable, so the PEP no longer applied. Those under the mistaken impression that the PEP is responsible for some mystery force might imagine the star would suddenly collapse. But what would actually happen is this: almost nothing! Globally and right away, anyway. That's because the pressure doesn't care if the PEP is active or not, it only cares about the momenta the particles have. If you wave away the PEP, all that would immediately happen is that the momenta and kinetic energy would start repartitioning into the more likely configuration of the Maxwell-Boltzmann distribution, with no effect at all on pressure. So much for "degeneracy pressure" versus "ideal gas pressure" as two different causes of pressure! But there would be a huge effect on temperature, revealing the fundamentally thermodynamic character of the PEP. So the energy transport would change drastically, and the white dwarf would become much brighter on a radiative diffusion time, but none of these changes would appear on the sound crossing time because degeneracy isn't a force.

 

[PeterDonis]

If you wave away the PEP, all that would immediately happen is that the momenta and kinetic energy would start repartitioning into the more likely configuration of the Maxwell-Boltzmann distribution, with no effect at all on pressure. So much for "degeneracy pressure" versus "ideal gas pressure" as two different causes of pressure! But there would be a huge effect on temperature, revealing the fundamentally thermodynamic character of the PEP. So the energy transport would change drastically, and the white dwarf would become much brighter on a radiative diffusion time, but none of these changes would appear on the sound crossing time because degeneracy isn't a force.

Let me try to rephrase this in more physical terms. The immediate effect of the magic wand is to open up a large number of low energy states for electrons that were not previously available (because of the PEP constraint). But that doesn't mean the electrons will instantaneously occupy those states; it just means the states are available. So there will have to be a process whereby electrons that are in higher energy states give up energy and occupy lower energy states. This process is a process of heat transport--the energy has to be radiated away from the white dwarf as a whole, out into empty space--and so its characteristic time scale is the radiative diffusion time, which is much longer than the sound crossing time. On the time scale of the sound crossing time, nothing significant will happen, because on that time scale not much energy has actually been given up by the electrons, so the pressure hasn't significantly changed. Once the energy has been given up, the pressure of the white dwarf will be much lower and it will have to contract significantly in order to re-establish a new equilibrium. (Actually this will have been happening as the energy was radiated away.)

Is this a fair summary of what you are saying?

(Btw, although the white dwarf will become brighter for a while, it will ultimately end up dimmer, won't it?)

 

[my2cts]

But what would actually happen is this: almost nothing!

What a mistake. I bet you don't have a reference for this to a real scientific paper or textbook.
What would happen is that all the extra energy that was required by PEP now becomes available as thermal energy.
The star would suddenly become much hotter, gravitationally implode under emission of huge amounts of radiation and explosively reject its outer parts. It would go supernova.
What would happen is the extreme opposite of "almost nothing".
This is what actually happens when a neutron star forms: the matter turns from degenerate into non-degenerate..

 

[PeterDonis]

This is what actually happens when a neutron star forms: the matter turns from degenerate into non-degenerate..

A neutron star is degenerate. So is the core of a star that goes supernova and forms a neutron star in the remnant. (These are different types of degeneracy--electrons in the star core vs. neutrons in the neutron star--but they're both degenerate.) I don't know what you're talking about here.

 

[my2cts]

A neutron star is degenerate. So is the core of a star that goes supernova and forms a neutron star in the remnant. (These are different types of degeneracy--electrons in the star core vs. neutrons in the neutron star--but they're both degenerate.) I don't know what you're talking about here.

It is too bad that you don't know what I am talking about, for this is really an interesting thought experiment, and on-topic as well..

When the electrons and protons are converted into neutrons the neutrons are not degenerate.
Indeed, a neutron gas at the same density as a degenerate electron gas plus nondegenerate protons
has a 2000 times smaller fermi energy that the electron gas, namely the ratio of neutron over electron mass.
It will have to shed this excess energy to become degenerate.
Its volume will shrink so it will also have to get rid of a lot of gravitational potential energy.
Boom. Supernova.
In the thought experiment of Ken G the end product will be a degenerate proton gas
embedded in a see of electrons that are no longer subject to PEP due to the "magic wand".
Such a body could be similar to a neutron star.

 

[Ken G]

What a mistake. I bet you don't have a reference for this to a real scientific paper or textbook.

So are you saying you dispute that P = 2/3 E/V for both degenerate and ideal gases? Goodness, I thought we had established that a long time ago. So if you realize that expression is correct, just ask yourself this: does the "magic wand" that makes the electrons distinguishable change the E, or the V, or neither? Of course, the answer is neither, so nothing happens to P at first. It's just obvious, it's very hard to find references for statements of basic math. Of course, I already said that things will begin to evolve over a radiative diffusion time.

What would happen is that all the extra energy that was required by PEP now becomes available as thermal energy.

What happens is just what I said-- nothing at all to the pressure, but a big jump in temperature.

The star would suddenly become much hotter, gravitationally implode under emission of huge amounts of radiation and explosively reject its outer parts.

If you reread my last post, you will find a more careful accounting of the timescales here.

It would go supernova.

Yes-- eventually. But not for quite a long time, and certainly not on a sound crossing time, as I said above. The star will need to wait for many radiative diffusion times in order to lose that energy that is now freed up. But this is all what I stressed above-- the primary effects of the PEP are on heat transport. You are only repeating what I already said, but exaggerating how quickly it would happen.

What would happen is the extreme opposite of "almost nothing".

Again, if you would quote me, it is important to quote my entire statement, or you will miss the important parts, as you have done here. You have missed the timescales involved.

 

[PeterDonis]

When the electrons and protons are converted into neutrons the neutrons are not degenerate.

Ok, that clarifies what you meant. I don't think it's correct, though. As I understand our current supernova model, or at least the one that's relevant for this discussion, gravity overwhelms pressure in the core of a star when nuclear fusion shuts down and the core's mass exceeds the Chandrasekhar limit. The star's core at that point is at white dwarf densities, not neutron star densities--i.e., the density is way too small to force inverse beta decay and convert electrons + protons into neutrons. It is true that the neutrons are not degenerate at that point; but that's irrelevant to the collapse process, because the collapse is not driven by the core radiating away excess heat from non-degenerate neutrons; it's driven by gravity overwhelming pressure. Electrons + protons turning into neutrons won't happen until the core has collapsed almost to neutron star densities--at which point the neutrons are degenerate.

In the thought experiment of Ken G the end product will be a degenerate proton gas
embedded in a see of electrons that are no longer subject to PEP due to the "magic wand".

No, it won't, because, by the same argument you gave for neutrons at white dwarf densities, the protons in the white dwarf just after the magic wand is waved are not degenerate.

Such a body could be similar to a neutron star.

No, it won't; see above. Nor will it collapse the way a supernova's core does, because the starting point was a stable white dwarf, which means its mass must have been below the Chandrasekhar limit, and therefore gravity does not overwhelm pressure. That means the key process going on in this case, unlike the core collapse supernova case, is heat transport.

 

[PeterDonis]

This subject has been sufficiently discussed. Thread closed.