Re: [Phys-l] equipartition riddle

[John Denker <jsd@av8n.com>]

On 04/09/2011 08:23 PM, Mike Viotti wrote:

> I'm afraid I don't see the riddle. 

OK, good point.  Let me make it more explicit.  I should have
formulated it this way the first time:  There is an objective,
easily-observable bit of physics here:

      We know that for N=2  the specific heat capacity is 1   
        in the appropriate units.
      We know that for N=∞  the specific heat capacity is 1/2 
        in the appropriate units.
Fill in the blank: for N=10 the specific heat capacity is ___ 
        in the appropriate units.

>  Doesn't it seem
> like a bit of a leap to say that the "2" coming from the x2 in the
> parabolic well corresponds directly to the 2 in the "degrees of freedom" of
> the particle in the well? 

I'm not sure what is meant by "directly".  Obviously the relationship
is not "direct" as in "directly proportional" since the specific heat
capacity goes down as N goes up ... and this has traditionally been
explained by counting degrees of freedom.

But somehow, indirectly, the observed specific heat capacity must 
be related to N.  If you don't want to count degrees of freedom, we
need to come up with another way of predicting and/or explaining 
the specific heat capacity.

> Where's the hangup?  Finding the degrees of
> freedom?  ...wouldn't it still be 2?

That's one way of attacking the riddle, if you so choose.
(The alternative is to attack the observable specific heat
capacity directly, without counting degrees of freedom.)

If there are two DoF for N=2 and N=10, where does the second
degree of freedom get lost?  Is it still there at N=1000?
Is it still there at  N=10000000?  We know it is not observed
at N=∞, so it evidently gets lost somewhere, somehow.  Does 
it fade out gradually, or does it vanish suddenly at some point?

Or maybe you could argue that the second DoF is still there but 
somehow doesn't contribute to the observable physics.

If you want to argue that the whole idea of "degrees of freedom" 
is bogus even in classical thermo, then we need to explain
why it has been so heavily used for so long, with so much
success (or apparent success?).  We know that the idea breaks 
down in the quantum regime, but the riddle is asking about the 
classical regime.

==========

This is IMHO not an easy riddle.  It bugged me for years and
years.  A few days ago I tripped headlong over the answer.