[Phys-l] independent variables, or not

[John Denker <jsd@av8n.com>]

On 01/13/2012 11:07 AM, LaMontagne, Bob wrote:
> Suppose you go to a local stream and measure the temperature and the
> speed of flow of the water. A plot of temperature vs time has an
> obvious dependent and independent variable - because you are in
> control of the time. Likewise for flow speed vs time. But suppose you
> decide to plot the two variables temperature and speed of flow
> against each other  - which is dependent - which is independent?
>
> Unless you have deliberate control over the choice of one of the
> variables, I don't really see where dependency and independency enter
> the choice of axes. I simply state to my students that the expression
> "plot A vs B" has a standard interpretation of A on the vertical axes
> and B on the horizontal.

Amen, brother.

This calls attention to an important issue, namely the very existence 
of "independent" versus "dependent" variables.  This is not entirely
unrelated to yesterday's discussion of which axes to use, but it is
not entirely the same, either.

As I see it:  *sometimes* you can identify independent and dependent
variables, but sometimes you can't.  Often you get to choose. Often 
it isn't worth the trouble of choosing, even if you could.

For example, many authors who ought to know better "define" certain
thermodynamic potentials to be functions that depend on certain
"natural" variables, to the exclusion of other dependencies:
   E(S, V)   energy
   H(S, P)   enthalpy
   F(T, V)   Helmholtz free energy
   G(T, P)   Gibbs free enthalpy

I wish they wouldn't do that,  It makes my job harder.

The most prosaic heat capacity experiment measures ΔT as a function 
of ΔE ... in defiance of the aforementioned "natural" variables.
I get kinda tired of students telling me it is impossible in 
principle to measure dT/dE because T "must" be the independent 
variable and E "must" be the dependent variable (and even then,
E "must" depend on S not T).  I don't care how many textbooks say
E must be E(S, V) ... Mother Nature doesn't care about such rules,
and the heat capacity measurement is conceptually and operationally
well-behaved, even though it violates these hare-brained "rules".

If you reverse course and decide that E "must" be the independent
variable, that's wrong, too.  In my house, I choose the temperature
via the thermostat, and the energy-content follows accordingly.
More generally, any feedback loop reverses the roles of a pair of
variables, converting dependent to independent and vice versa ...
if you wish to think in such terms at all, which I don't generally
recommend.  There are lots of feedback loops in the world.



As another example, consider the inverse temperature, β = ∂S/∂E
at constant V.  There are good physics reasons to prefer β rather 
than T ... especially in spin systems near β=0, in which case β 
is perfectly well behaved, in contrast to T which has all sorts 
of problems (discontinuous, undefined, et cetera).  I get kinda 
disappointed when students freak out over ∂S/∂E, on the grounds 
that E "must" be the dependent variable and S "must" be the 
independent variable.


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

Here's the smart way to think about it:

By way of analogy, consider a vector as defined by physics, perhaps
the velocity vector of some particle.  This vector is visualized in
terms of "tip and tail".  It exists in terms of physics *independent* 
of what coordinate system -- if any -- we choose to use.  We might
use the red coordinate system, or the green system (which is rotated 
relative to the red system) or the blue system (which is moving
relative to the red system) ... or none of the above.  It's the same
vector, no matter what.

Similarly, in thermodynamic state-space, think of the state as existing 
in terms of the physics, independent of whatever coordinates -- if any -- 
we wish to impose on it.  A small displacement of the state is a vector
in this space (or rather in the tangent space, if you want to get fussy).
The state-displacement vector has a tip and a tail.  You can parameterize
the displacement in terms of some abstract parameter λ, such that λ=0 is
the tail and λ=1 is the tip.

Here's the payoff:  If you insist on thinking in terms of independent
and dependent variables, λ is the independent variable and everything
else is dependent.  That means E, H, F, G, T, S, P, V, and all the rest
are functions of λ.  You can draw pictures of this and visualize it in
terms of the topology of thermodynamics state-space.

Using these ideas, we can write things like

             dS / dλ
        β = ----------      when dV / dλ is zero
             dE / dλ

This parametric approach is useful in a wide range of situations.  It
is verrry common to plot something that is not a function of either
axis-variable, but is instead a function of some parameter.  Yesterday
I mentioned the power-curve example and the Archimedes-spiral example.


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

Tangential remark:  Of course there are situations where you might be 
interested in E(S,V), H(S,P), F(T,V), and/or G(T,P) ... for instance 
if you are interested in questions of stability, spontaneity, and 
reversibility ... but there is no law that says you have to be interested 
in such things to the exclusion of other things, such as heat capacity. 
For more on stability, spontaneity, and reversibility, see
  http://www.av8n.com/physics/spontaneous.htm