Re: [Phys-l] equipartition riddle

[Mike Viotti <mike.viotti@gmail.com>]

John,

I'm afraid I don't see the riddle.  If the potential goes as x^10, then the
potential goes as x^10.  Where's the hangup?  Finding the degrees of
freedom?  ...wouldn't it still be 2?

Granted it has been a while since I've considered a lot of this, and I
follow steps (a)-(c), but it doesn't seem like we're really generalizing
anything, so my mind isn't exactly blown with step (d).  Doesn't it seem
like a bit of a leap to say that the "2" coming from the x^2 in the
parabolic well corresponds directly to the 2 in the "degrees of freedom" of
the particle in the well?  Maybe I'm missing the riddle.  Could you lay it
out a bit more explicitly for me (off-list if necessary, to maintain the
riddle for those who are more in the know)?

Mike, looking forward to having his mind blown.

On Sat, Apr 9, 2011 at 7:39 PM, John Denker <jsd@av8n.com> wrote:

> Here's a scenario ... with a riddle in part (d).
>
> a) Suppose we have a harmonic oscillator i.e. a particle in a parabolic
> potential well.  It is in thermal equilibrium at temperature T.  We
> are interested in the classical limit, i.e. where kT is high compared
> to ℏω.  In this case the average energy is kT i.e. 1/2 kT per degree
> of freedom, where the PE is one degree of freedom and the KE is another.
>
> b) Same as above, except that the particle is in a square well potential.
> In this case the average energy is 1/2 kT, not kT.  Apparently in this
> case the KE counts as a degree of freedom but the PE does not.
>
> c) By way of formalism, note that we can consider the two previous cases
> together, as follows:  We say the potential goes like x^N, where in
> case (a) we have N = 2, and in case (b) we have N = ∞.
>
> d) So the question arises, what about the intermediate case?  What
> happens in the case where the potential goes like x^10, where 10 is
> bigger than 2, but a lot less than infinity?
>
> How do we interpolate between two degrees of freedom and one degree
> of freedom?
>
> Note: This riddle bugged me for many many years.  I learned the answer
> a few days ago.
>
> ===========
>
> Give-away:  Sturge Appendix G.
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