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Re: [Phys-l] how not to do thermodynamics and Legendre transformations

In the context of
R.A. Alberty
http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf

On 10/13/09 10:55, Carl Mungan wrote:

Interesting that you mention that paper, because a chemist friend
recently cited that exact paper to me and told me that he was still
confused about Legendre transforms and could I explain it to him more
simply?

In reply I wrote:

http://usna.edu/Users/physics/mungan/Scholarship/LegendreTransform.pdf

and would glady accept your (or other folks) comments on it.

Some possibly constructive suggestions:

1) Instead of expressions such as

∂f
u ≡ ---- [1a]
∂x

∂f
w ≡ ---- [1b]
∂y

I would strongly recommend writing the more-explicit expressions

∂f |
u := ----| [2a]
∂x |y

∂f |
w := ----| [2b]
∂y |x

or perhaps the more compact versions:

∂f
u := ------ [3a]
∂x|y

∂f
w := ------ [3b]
∂y|x

The point is that partial derivatives are like directional
derivatives. In expression [1a] the denominator indicates
we are taking a step that changes x, but alas in the
context of thermodynamics there are innumerable inequivalent
ways of doing that. We need to specify that we are taking
a step that changes x _in the direction of constant y_.

We all know there is tremendous temptation to simplify the
notation by leaving out the "at constant y" qualifiers,
and sometimes you can get away with this, if x and y are
known to be the only variables in town ... but there are
plenty of cases where x and y are not the only variables.
Specifically, one of the things that makes thermodynamics
hard is the necessity of dealing with multiple non-independent
variables.

I find it pays to be super-explicit about the direction
of each partial derivative, *especially* when
-- dealing with students and/or
-- dealing with somebody who self-identifies as being
already confused.

It pays to set a good example. This includes making
the direction explicit even if you personally know the
direction, because the other guy might not know the
direction. Conversely, there is no need to teach
students how to simplify the notation; they will do
(and overdo) that all on their own.

===================

2) As for the new potential g := f - wy, I would not
have said that x and w are "its independent variables".
The idea that certain potentials "possess" certain
variables is unfounded and unhelpful. The way I see
it, f and g and x and y and u and w exist at every
point in thermodynamic space. Today you may choose
to differentiate f with respect to (x,y), but tomorrow
you may wish to choose differently.

Thermodynamic space doesn't have any geometry, but it
does have topology ... including differential topology.
To repeat: there is no metric in thermodynamics space,
hence no usable notion of lengths or angles. The
beautiful thing about differential topology is that
you can write things like equation [3] *without* any
requirement for x and y to be orthogonal. There is a
perfectly good topological interpretation to taking a
step that changes x along a contour of constant y (as
in [3a]) or taking a step that changes y along a contour
of constant x (as in [3b]).

Just to be super explicit, it is perfectly OK -- and
sometimes very useful -- to write dg in terms of dx
and dy not dw:
∂g ∂g
dg = ------ dx + ------ dy [4]
∂x|y ∂y|x

As a concrete example, it just cracks me up when books
insist that the energy E is a function of (S,V) but then
concentrate on heat capacity Cv. We can understand Cv
in terms of:
∂E ∂E
dE = ------ dT + ------ dV [5a]
∂T|V ∂V|T

∂E
= Cv dT + ------ dV [5b]
∂V|T

The potential g does not "belong" to (x,w) or vice versa.
The potential g is not "naturally" associated with (x,w)
or vice versa. The energy E is not "naturally" associated
with (S,V) or vice versa. There is nothing to be lost
and much to be gained by flushing the whole notion of
"natural variables" and relying on the topological and
pictorial interpretation of what the derivatives mean.
-- You can picture what's going on.
-- Students can picture what's going on. Draw the
contours of constant x and the contours of constant y.
They do not need to be orthogonal. Take a step that
changes x along a contour of constant y or vice versa.
-- This is not rocket surgery. The smart way of doing
things is _easier_ than the dumb way. It's also more
powerful.

As the saying goes, it's dumb to eat vichyssoise with
a fork, when spoons are readily available.

2b) Minor supporting point: I think of E as a function
_state_ directly, and only very indirectly as a function
of (S,V) or (T,V) or any other variables. By the same
token, F, G, and H are functions of state. Forsooth,
S, T, P, V, et cetera are functions of state. They exist
at each point in state space.

I find it super-helpful to clearly _picture_ what the
derivatives mean. Pictures work together with the
equations. Neither substitutes for the other; both
are needed.

===================

3) By way of motivation, it sometimes helps to mention
the connection between Legendre transformations and
integration by parts. It makes the Legendre transformation
seem less mysterious. Most people find it relatively
easy to visualize the meaning of integration by parts.

===================

4) A much much smaller point about the notation: Comparing
equation [2] to equation [1] you see that I prefer defining
things using the := operator rather than the ≡ operator.
-- Equalities are symmetric: a ≡ b and vice versa
-- Definitions are not symmetric: a is defined in terms of
b and not vice versa. I like to use an asymmetrical
symbol for an asymmetrical idea.