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One context where this comes up is the equipartition theorem.
I don't want to write simply U = fNkT/2 (where f is the number
of degrees of freedom per particle), because most systems
also contain other energy, such as energies in chemical bonds
and rest energies of all the particles. So instead I write
U_thermal = fNkT/2, with the understanding that under a limited
range of conditions, the change in U is the same as the change
in U_thermal. . . .
My question for you, Leigh: How do you write the equipartition
theorem as an equation that can be applied to an entire physical
system (as opposed to a single abstract "degree of freedom")?
I haven't a clue. Since the equipartition theorem applies to a model
with quadratic degrees of freedom I guess I would say that the system
internal energy has equal magnitude contributions associated with
each quadratic degree of freedom, together with the caveat that one
must be far from the temperature where quantum phenomena are
important.
There's still lots of room for other terms. Remember, I'm teaching a
course in astrophysics right now. We deal with stars. There are lots
of non-quadratic energy terms in a plasma.