Re: "non-transfer" of energy

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's comments:
> ...
> 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.

It appears that although John Denker and I have philosophically
similar views on the concept of flowing stuff, they are not precisely
the same at the level of the actual definition of the concept.

John conceives of a 'flow' without a 'fluid' field.  I don't. John is
happy to consider that an isolated point particle (with a delta
function density) can flow; I am not.  From my perspective I am happy
to allow that such a particle can move along a continuous (in time)
trajectory, but I would not consider it to be a flow when it does. To
me the concept of flowing necessarily entails a concept of a
continuous density field for the flowing stuff.  Delta distributions
are just too lumpy for me (but not apparently, for John).  The way I
see it 'flowing' is a form of continuous-in-time motion, but not all
forms of continuous-in-time motion are necessarily a 'flow'.  I want
a continuity in *space*, too.  Trajectories of point particles are
too concentrated in space to qualify according to the notion I have
in mind for what it means to flow.  I want a spatially continuous
density so that there is a continuity in the motions of neighboring
infinitesimal parcels.  Isolated delta functions do not have this
property.  That is why I wanted the infinite divisibility requirement
to hold at the *macroscopic* level of flow description so that *even*
if the 'fluid' *was* made of classical point particles at the
microscopic level, at the macroscopic level of the fluid description
the density would be still averaged over a sufficiently large course-
grained scale size (that is still very small compared to the
macroscopic scale size) that the resulting density would be
sufficiently continuous in space to be able to flow (according to my
lights).

I could accept a flow for a single particle if the particle happened
to be a quantum particle (possibly even in a single pure quantum
state) and the flow was for the probability density and flux for the
particle's position in space.  The Heisenberg Uncertainty Principle
prevents such a 'probability fluid' from being fully concentrated
into a classical trajectory of a moving delta function.

In short, for me, 'flowing' requires some sort of mathematically
spatially continuous 'fluid' field to be the density of the 'flowing
stuff'.  For John, he is happy to have a 'flow' without the
supporting 'fluid' concept.  For me 'fluids' 'flow' (where the
concept of a 'fluid' entails a spatial and temporal continuity in
its density & flux current); non-fluids don't.

> 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.

True.  But I wouldn't consider a tossed baby to 'flow' either. Move,
yes, flow, no.  The tossed bath water containing the baby *would*
flow, though.

> 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.

But for me, if such strategems are necessary, I doubt that I would
call the motion a flow.  I want the effectively infinite divisibility
requirement so the flowing density field can be spatially continuous.

> ===========
>
> 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.

Charge is extensive.  Its density is not.  The flow is a flow of
charge (that is continously smeared into a finite density in any
effective region of space large enough so that the quantization of
charge is not observable) from one region of space to another one.
The 'flowing fluid' has a spatially and temporally continuous
density, but it is *itself* an extensive cumulative property whose
total value in any region of space is the sum of its values in any
partition of that region into subregions.  As a counterexample,
consider a temperature field for some continuous hydrodynamic matter.
Such a field does not have the property that the spatial integral of
the temperature over some region of space represents a total
accumulated amount of any relevant physical property.  What I want is
that the spatial integral of the continuous intensive density
actually represent some relevant extensive physical property of the
system.  I want that the density field actually be a density of some
sort of extensive spatially cumulative property.  The 'flow' is then
a continuous (in both space and time) transport of this extensive
property from one region to another.

Since I don't conceive of the color blue as being extensive in space
I don't consider total blueness to be able to flow--even though a
blueness property may be present and carried by large enough
macroscopic groups of particles that *can* flow (because the number
of those particles *is* cumulative in space, and such macroscopic
groups of particles can be effectively infinitely divisible).

When the 'flowing fluid' is *extensive* then a flow of it from one
region to another will tend to result in an increase in the
accumulated amount of it in the region it flowed into and a
corresponding decrease in the accumulated amount of it in the region
it flowed out of.  For me, having this cumulative extensiveness in
the 'flowing' property is necessary for there to be a 'flow'.

> ==========
>
> Regarding locality:  this is definitely a good
> point.

I'm glad we agree on this 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.

And being able to flow is not one of them.

David Bowman