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[Phys-l] thermodynamic dot products, or not

On 01/11/2010 01:04 PM, Stefan Jeglinski wrote:

dE = F dot dx [1]

This to me is, or I want it to be (grin), the great unifying concept,
but I'm not sure how to word the concept succinctly or also determine
whether I can always sleuth it in a physics problem.


We are all
familiar with F dot dx from mechanics, and using it to explicitly
write down the mechanical potential or kinetic energy. But it seems
also to be highly general, a way to tie thermo and mechanics
together, dE being a sum of as many terms "as you can think of that
apply to the problem." The dot notation reinforces the gradient
notion JD brought up. Eschewing signs, we have terms such as T dot dS
(entropy), P dot dV (mechanical, just another form of mechanics' F
dot dx?), mu dot dN (particle exchange), and the aforementioned V dot
dQ (charge transfer). To be conservative, the general term "F" must
itself be the gradient of a potential, no? So we have 2 gradient in
what I have called a sum of generalized F dot dx terms?

OK, so now we are considering generalizations of
equation [1]. A simple example is

dE = T dS - F dot dx [2]

where I have switched to a slightly more conventional sign

That equation can be expanded as

dE = T dS - F1 dx1 - F2 dx2 - F3 dx3 [3]

where [F1,F2,F3] are the components of the three-dimensional
vector F.

So if I understand the question correctly, the question has
two parts:
a) whether [T,F1,F2,F3] is some sort of four-dimensional
vector in thermodynamical state space, and
b) whether the RHS of equation [3] is a dot product
involving this sort of vector.

The answers are:
a) Yes.
b) No.

There are vectors in state space, but no dot product.

Point (b) comes as a shock to some people ... but it's
true and important. Thermodynamic state space has a
topology but no geometry. There is no metric, no dot
product, no notion of length, and no notion of angle.
There is no sense in which [T,0,0,0] is perpendicular
to [0,F1,0,0] or to anything else.

We need to respond to this in a nuanced way:

-- On the one hand, we need to take this seriously.
We need to be careful. Thermodynamic state space
is different from ordinary XYZ space, different in
important ways. There are things you can get away
with in XYZ space that you cannot get away with in
thermodynamic state space. This includes dot products.
This includes any notion of the P direction being
perpendicular to the V direction. Students get
this wrong all the time, and it messes them up.
There is negative transference from XYZ space to
state space.

-- On the other hand, there are still lots of useful
things we can do in state space. Even though we
cannot identify any particular direction as the E
axis, we can identify contours of constant E, and
that's all we really need in order to do calculus,
up to and including Taylor series.

There is a name for this, for the idea of doing calculus
in a space that has a topology but not a geometry. Not
surprisingly, it's called _differential topology_. It's
a rather new branch of mathematics; it didn't really get
going until the 20th century. It is heavily used in some
branches of modern physics ... but for some reason it has
barely begun to seep into the thermodynamics texts. I
heard about the topology of thermodynamics in grad school
... from one of the students. The professors who were
teaching the thermo course had no clue about it.

There are big fat books on the subject of differential
topology ... but you do not need to master whole subject
in order to do useful things with it. A short, relatively
simple summary of the bits you need to know is here:

I find it important to _visualize_ what's going on in
state space. For example, I visualize dE as a vector,
not a pointy vector but rather a set of contour lines.
This is a huge improvement over trying to think of dE
as an infinitesimal, which is both mathematically unsound
and impossible for me to visualize.