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*From*: John Denker <jsd@av8n.com>*Date*: Tue, 12 Jan 2010 05:44:01 -0700

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.

OK.

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

convention.

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:

http://www.av8n.com/physics/thermo-forms.htm

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.

**Follow-Ups**:**Re: [Phys-l] thermodynamic dot products, or not***From:*"Rauber, Joel" <Joel.Rauber@SDSTATE.EDU>

**References**:**[Phys-l] T dS versus dQ***From:*John Denker <jsd@av8n.com>

**[Phys-l] thermodynamics of dissipation***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermodynamics of dissipation***From:*Stefan Jeglinski <jeglin@4pi.com>

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