[Phys-l] grad of potential (was: configurational energy)

[John Denker <jsd@av8n.com>]

On 10/17/2007 03:05 PM, Jacques Rutschmann wrote:

> > 4) In physics and related disciplines, "potential" means 
> > positional, and has been used in this sense more-or-less
> > consistently since about 1840 (or perhaps earlier;  I don't know).
> > This extends to gravitational potential, electrostatic potential,
> > et cetera.  By a further extension, any functions of state are
> > called thermodynamic potentials e.g. chemical potential, Gibbs
> > potential, et cetera (which exist in an abstract state-space, not
> > in prosaic position-space).

> This is helpful!

:-)

>  I always believed that "potential" meant in physics
> that the force (or more abstract quantities) could be computed by
> taking the grad (*) of the "potential"... I always found very
> disturbing that the gradients of the numerous thermodynamics
> potentials were of no interest. 

We probably agree 100% as to the sentiment, but you have to
be careful how you express it.  Cases to consider include:

a) In an inhomogeneous fluid, the derivatives (w.r.t position)
of a thermodynamic potential such as energy per mole (E) might 
be of considerable interest.  If we know E as a function of x 
and y we can write:

             ∂E          ∂E
     d(E) = ---- d(x) + ---- d(y)                        [1]
             ∂x          ∂y

b) In a homogeneous fluid, that's not so interesting, because
the derivatives (w.r.t position) vanish.

c) At a given x,y point in the fluid (homogeneous or otherwise)
we might know E as a function of V and S, in which case we can
write:

             ∂E          ∂E
     d(E) = ---- d(V) + ---- d(S)     at constant x,y    [2]
             ∂V          ∂S

which might be very, very useful.

Both [1] and [2] have an equal claim to be "the" gradient of E.
They are similar in form, and profoundly similar in meaning.

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

This can be made very precise at almost zero cost if the d(...)
operator is interpreted as the exterior derivative.  That makes
it a one-form.  One-forms are best visualized in terms of contour 
lines, as on a topographic map.  They are vectors, since they
uphold the axioms that define a vector space.  They are not,
however, equivalent to the more familiar pointy vectors.

Almost anything you really need to do with a gradient vector can
be done with a gradient-one-form-vector;  you don't need a 
gradient-pointy-vector.

In particular, in abstract thermodynamic state-space, there is a 
gradient-one-form-vector, but there is no gradient-pointy-vector.  
State space has no geometry, but it does have a differential 
topology.

This is worked out in detail, with illustrations, at:
   http://www.av8n.com/physics/thermo-forms.htm