Re: thermal energy

[Daniel Schroeder <DSCHROEDER@CC.WEBER.EDU>]

> Two related comments.  I think that limiting the equipartition theorem to
> quadratic degrees of freedom is rather err, well "limiting".
>
> I think one can extend the theorem to other sorts of degrees of freedom, of
> course it no longer would be kT/2 per degree of freedom, but would have to
> be changed to kT/2 for the quadratic guys and #### for the degrees of
> freedom of type A and $$$$ for the one's of type B, etc etc.
>
> And for Dan's comment, aren't some of those internal chemical energy degrees
> of freedom quadratic in nature and included in the equipartition theorem.
> E.g. any chemical bond that is reasonably modeled as a Hooke's Law spring
> and therefore having vibrational kinetic and potential energy degrees of
> freedom?
>
> Joel Rauber

You can certainly generalize the equipartition theorem to other
power-law energy functions.  I'd be interested to hear of generalizations
to non-power-law functions if anyone out there knows of any.

But as a practical matter, in the common, everyday situations it's
the quadratic energy functions that lend themselves to a simple
treatment.  Look through any table of heat capacities of solids
or gases (let's leave liquids out of it).  You can explain the
vast majority of them, at least approximately, using the
equipartition theorem for quadratic degrees of freedom,
U_thermal = fNkT/2.  So even though this "theorem", as I've
stated it, requires all sorts of caveats, in my mind it works
too well to ignore.

Dan