Re: heat is a form of energy

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding where Leigh wrote:
> ...
> I understand the energy to be a state function ascribable to an
> isolated (or hypothetically isolateable) physical system.

So far so good.

> It is exactly the same sort of state function as the entropy, but
> of course it varies in different ways as the universe evolves.
> If there were no such differences between energy and entropy it
> would be unnecessary to formulate the two quantities. You have
> not considered the great difficulty students have in learning
> the simple meaning of the statement that entropy is a function
> of state, an abstract thing that is utterly insubstantial -
> *just like energy*.

I tend to disagree with the thrust of this.  I do not see energy as being
*at all* similar to entropy.  Energy is a function of the *microscopic*
state of the isolated system.  This statement is correct as it stands for
a classical system.  For a quantum system the energy is strictly
definable as a function of the microstate only if that microstate is an
eigenstate of the Hamiltonian.  If the microscopic quantum state of the
quantum system is a superposition of various energy states then the exact
energy value is not quite determined, but the energy still has a well
defined mean expectation value anyway.  If the quantum system is
macroscopic in the number of its internal degrees of freedom, then for
any realizable state the distribution of quantum energy eigenvalues for
the microstate is *very* narrow relative to any experimentally applicable
scale of energy resolution of measurement.  Such a system effectively has
a well defined energy as a function of its microstate anyway.  Thus,
either in the classical or quantum system the system's energy is
essentially a function of the system's microstate.

The entropy is not like this at all.  The entropy is a function *not* of
the individual actual microstate, but is a function of the probability
distribution of possible microscopic states consistent with the
macrostate.  Since we typically have that a given macroscopic description
(macroscopic state) determines the distribution of possible microscopic
states consistent with that macroscopic state (such as in the case of a
thermodynamic equilibrium characterized by a handful of important
macroscopic parameters--one of which is the internal energy), we see that
the entropy is a function of the *macroscopic* state.  The entropy
doesn't have a meaningful value for a given microscopic state.  If
anything, the entropy of the given microscopic state is zero since there
is only 1 microscopic state that the system can be in when is in that
particular microscopic state of interest, and log(1) = 0.  Being a
function of a probability distribution, which itself is a function of the
macrostate makes the entropy an intrinsically statistical concept, that
is also a function of the macrostate as well.  This is very different
from the concept of the energy which is merely the value of the
Hamiltonian evaluated for the microstate that the system is actually in.
Since we do not have microscopic precision in evaluating the energy we
take the macroscopic energy to be the average of the (macroscopically
close by) energies of the individual microscopic states averaged over the
distribution of those possible microstates where the standard deviation
of the distribution of possible energy values is quite negligible
compared to the mean value because the statistical fluctuations in the
energy tend to be much too small to show up at the macroscopic level of
description.

> Treating the energy as being substantial is the great cognitive
> block to the student's later understanding of the entropy.

I think *any* concept of energy--substantial or not, ought not influence
one's concept of entropy, which, at base, has *nothing* to do with energy
per se.

> Some
> teachers even try to smooth this over by introducing the "flow
> of entropy" as a helpful concept!

This would not help any understanding of entropy.  Fortunately, I have
not encountered this teaching technique.

> I apologize for my lack of ability in articulating what is to
> me a simple conceptual framework. I do recognize that "reality"
> is a property which is properly discussed by metaphysicists
> rather than physicists, but I have never been a terribly proper
> fellow.

No need to apologize here.  If we didn't have different metaphysical
perspectives, we would have a lot less to discuss on this list--even if
is supposed to be for 'just' physics.

David Bowman
dbowman@georgetowncollege.edu