Generalized Flow. was (Re: Energy)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's comments:

> Some field lines terminate on charges, so there is a conservation
> law here:  Write the good old conservation-of-charge law and
> differentiate both sides.  Other field lines don't terminate at all;
> they just chase their tails endlessly.  (Such field lines are
> created by time-varying magnetic fields.)  The corresponding
> displacement currents obey the law of conservation of zero.  Add
> these two together and I think you've got Joel's conservation law.
>
> I don't think the new law tells us anything beyond what the old
> conservation-of-charge law tells us.

I'm not sure about this 'conservation of zero thing' and whether or
not it is compatible with my reasoning about the non-flowing nature
of 'generalized' (i.e. 'actual' + 'displacement') currents.  So let
me explain my reasoning a little further.

Consider some locally conserved continuously extensive quantity Q.
Let's define q to be the local density of Q and let j_q to be the
flux current density of this Q.  Then, by hypothesis, we have the
continuity equation:

0 = dq/dt + div(j_q)

where the time derivative is taken to be a partial derivative.
This means that since dq/dt is also a divergence (note that -dq/dt
is explicitly the divergence of at least j_q among an infinite
assortment of other possible fields that could be chosen), let's
pick one such field, let's call it j_d, whose divergence happens to
be dq/dt.  If desired, we could even construct a suitable j_d by a
direct integration of dq/dt with some subsidiary conditions, or by
maybe solving a Poisson equation whose source field was dq/dt and
then taking the gradient of a scalar potential field that solved
it.  Once we have a suitable j_d (which is our displacement-like
term) we substitute it in for dq/dt in the above continuity
equation and get:

0 = div(j_d) + div(j_q) = div( j_d + j_q)

Next we define the 'generalized current flux'density:
j_g == j_d + j_q and then write the equation:

0 = div(j_g)

In general j_g is not the zero function.  Its precise value
depends on just which j_d was chosen whose divergence gave
dq/dt. It is true that this flux current density j_g does
satisfy the continuity equation:

0 = dg/dt + div(j_g)

where the 'generalized' density g is just the zero function and
the total 'generalized conserved charge' G giving that (trivial
& degenerate) function is zero.  But if the density function is
exactly the zero function I would not consider the resulting
'continuity equation' to describe an actual flow of some actual
quantity G (whose density if g).  In order for this
interpretation to hold up we would have to imagine a flow of
zero, i.e. G, from place to place, yet with a (generally)
nonzero current density j_g which just happens to always be
transverse.  This strikes me as perverse.

It seems to me that it is much better to simply admit our
'generalized current' density j_g is merely a transverse field
whose field lines always must "chase their tail endlessly".
If we integrate div(j_g) over any finite volume of space we
find that the net flux of j_q (i.e. the surface integral of
the normal component of j_q) out of (or into) that region,
(i.e. a "node") is zero.  Since there is no *nonzero* conserved
extensive quantity G whose density g is (at least somewhere) a
nontrivial density function obeying a continuity equation, I
would say that this does not describe a flow-type process in
general since there is nothing being accumulated or depleted
anywhere.  There is no change in the value of any G anywhere in
any finite region of space.  If nothing changes, it's kind of
silly to say that that 'nothing' is flowing(--especially with a
nonzero current density j_g to boot).

David Bowman
David_Bowman@georgetowncollege.edu