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.