Re: "non-transfer" of energy

["John S. Denker" <jsd@MONMOUTH.COM>]

David Bowman wrote:

> ... I would only add that in order for such a local continuity
> to exist (so that a continuity equation can describe it) a few other
> properties need to be satisfied first.
>
> Being able to flow seems to require about 4 prior conditions, and
> none of them necessarily have anything to do with the 'fluid' being
> any kind of actual substance.  First, the prospective quantity needs
> to be effectively infinitely divisible in space at the macroscopic
> level (so that macroscopic differential relationships can apply to
> macroscopically infinitesimal-sized regions that are themselves
> effectively so large in size on a microscopic level that they are
> already in the thermodynamic limit.  Second, the quantity needs to be
> *extensive* at the level of the macroscopically infinitesimal regions
> defined in the first point.  Third, the value of the quantity has to
> be capable of changing from place to place from time to time.  And
> the fourth condition is what I call a 'locality of causation'
> condition.

I mostly agree with that, but I have a couple of
comments and questions:

First, regarding the divisibility requirement:
Although it sounds like a serious limitation, and
although it is sometimes serious, it is often not
as serious as you might at first think.

For starters, consider a fluid of point particles.
Anywhere you find a particle, there is a delta
function of density there.  And if the particle
moves, it makes a delta function of current.  A
collection of such particles obeys the letter and
the spirit of the local conservation law, so we
can say that it flows.  It halfway meets David's
divisibility requirement:  on large length-scales
(the hydrodynamic limit) you have a fluid that
might be locally uniform and/or homogeneous, and
you can certainly subdivide it, but if you divide
it finely enough it will lose all semblance of
uniformity.  Even though it is flowing you might
not want to call it a fluid, because it loses
basic fluid properties like local velocity,
density, and pressure.

To repeat:  I'm perfectly happy talking about a
single isolated point-particle flowing from one
region to the next.  But I wouldn't call it a
fluid.

There are other ways in which divisibililty can
become problematic, and various ways of dealing
with it.  A baby, for instance, is proverbially
hard to divide.  But still, babies obey a
conservation law with only very infrequent
source and sink terms.

In sports, they don't award "half" of a first
down if half of the football crosses the
boundary of the "control volume".  Instead,
the arbitary rule is that if any part of the
football crosses the boundary, it's a first
down.  Similar strategems can often be used
to deal with situations where we can neither
subdivide things nor take averages.

===========

I don't understand the "extensive" requirement.
Electric charge conservation can be expressed
in terms of extensive Q and I, but it can also
be expressed in terms of intensive rho and j,
and I prefer the latter, because it emphasizes
the truly local nature of the conservation.
Any intensive quantity can be integrated to
yield an extensive quantity ... but if there's
a deep important insight here I'm missing the
point.

==========

Regarding locality:  this is definitely a good
point.  Perhaps a counterexamaple will highlight
it.  Ask what is the fastest-moving particle in
the control volume.  If you double the size of
the control volume, you might pick up a faster
particle, or you might not.  This quantity is
neither exensive nor intensive, and it is not
local.  It's well defined, but it isn't conserved
and has all sorts of peculiar properties.