Re: [Phys-l] thermodynamic dot products, or not

[John Denker <jsd@av8n.com>]

On 01/12/2010 06:44 AM, Rauber, Joel asked:

> If one can identify curves of constant E (or whatever coordinate),
> doesn't the idea of a tangent to the curve imply some sort of notion
> of directionality?

That's a good, incisive question.  

Here's the deal:

Remember that except in trivial cases, thermodynamics 
is highly multi-dimensional ... and the set of 
interesting variables is overcomplete.  As a simple 
example, suppose we have a three dimensional state-
space and five variables (C, D, E, F, G).  Then the 
contour of constant E is a two-dimensional shell, 
not a "curve" strictly speaking.

Now suppose we pick two variables, E and F.  Then dE
represents the shells of constant E and dF represents
the shells of constant F.  The intersection dE /\ dF
suffices to define a good "notion of directionality". 
It is the direction of constant E and F.

Note that the direction of constant E and F cannot be
considered the "E axis" or the "E direction", since
usually we think of the E axis as "the" direction in
which E is changing ... and by construction E is not
changing at all along the dE /\ dF direction.  Similarly
the dE /\ dF direction cannot be considered the C, D,
F, or G direction.

If we had only three variables, we could *maybe*
think in terms of a direction where "everything" is
constant except E.  But that is a bad idea in general,
and fails miserably when the set of variables is
overcomplete.  Some days you want to let E change
along the dC /\ dD direction, sometimes along the
dC /\ dF direction, et cetera;  there is no good
way to decide what is "the" direction in which E
changes.
 
> I suppose I could paraphrase by asking if tangent spaces exist for
> the Thermodynamic State Space?

Sure, a tangent space exists at each point.

But there's no dot product in the tangent space.