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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.
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Give-away: Sturge Appendix G.