Re: Whence Degeneracy Pressure?

["John S. Denker" <jsd@MONMOUTH.COM>]

At 10:46 AM 1/27/02 -0800, John Mallinckrodt wrote:
...
> Why should the Pauli principle *not* be considered itself to be
> the equivalent of a "fifth force"?

Because it's not.

> Most sources explain the origin of degeneracy pressure along
> something like these lines:  Fermions cannot occupy the same
> quantum state.  Thus, as the spatial extent of their wavefunctions
> become more and more similar, they are forced to occupy higher
> energy levels.

So far so good.

>  This is equivalent to the effects of a repulsive
> force.

Not "equivalent".  Similar, maybe, but not equivalent.

A spring inside a piston+cylinder is similar to an ideal
gas inside a piston+cylinder, but it is not equivalent.
In particular, the thermodynamics is radically different.

> ... the effect is at least *seemingly* independent of
> whether or not the particles interact in any other way--i.e., via
> the strong, electroweak, or gravitational mechanisms.

That's true.

> Now it is
> true that all fermions *do* interact by at least one of these
> mechanisms, but *is* this a requirement for fermions?

Well, they're pretty much required to interact with the gravitational
field, but depending on context this may or may not be an important
requirement.

I assume these fermions interact with their container, even if they don't
interact appreciably with each other.  Otherwise how would you measure the
so-called degeneracy pressure?

> ... if not, how would we explain the existence of
> degeneracy pressure in the case of two noninteracting fermions?

When people talk about discovering the Nth force, they are usually talking
about discovering something on a par with the canonical fundamental forces:
 -- gravitation
 -- the electroweak interaction
 -- the strong interaction.

What about, say, the lift on an airplane wing?  That's force.  But we don't
consider it a fundamental force, because it can be explained in terms of
the canonical fundamental forces.  It's a somewhat lengthy explanation, but
we have absolutely no reason to believe that anything fundamentally new is
going on.

What about, say, the force I feel when I grab a gyroscope and forcibly
re-orient its axis?  This is actually a good analogy to degeneracy
pressure, because it is _not_ particularly dependent on the gravitational
potential, nor the electroweak potential, nor the strong potential.  It is
sensitive to the kinetic energy.  Kinetic, not potential.  It is so readily
explained in terms of old-fashioned momentum etc. that it would be
confusingly redundant to add it to the list of fundamental forces.

So it is with the so-called degeneracy pressure.  It is associated with a
PdV energy (pressure times change in volume).  This energy is _not_
associated with the gravitational potential, nor the electroweak potential,
nor the strong potential.  It is a kinetic energy.

Just as you can have an ideal classical gas, or an ideal Bose gas, you can
have an ideal Fermi gas.  The particles are assumed to interact with the
container, but not significantly with each other.  In this ideal case, all
the pressure is associated with the kinetic energy of the gas.  When things
become degenerate, the Bose gas has a lower pressure than what you might
have expected from extrapolating the corresponding classical gas, while the
Fermi gas has a higher pressure.  But nothing has really changed.  The
"degeneracy" pressure is just pressure.  No new fundamental forces are
involved.