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Re: max entropy - more discussion



Joel continued the max. entropy discussion with:

This is admittedly a more minor part of the discussion, but here goes:
The pressure is an average quantity, it is averaged over something.
Speaking from an operational viewpoint; I'd say my pressure guage averages
over time.

Operationally, it averages over both time *and* the area of the sensing
surface.

And I don't think I'm willing to say the fluctuations my
sensitive guage sees is not registering fluctuations in pressure. I think
that saying it isn't, and is rather measuring something about the
microscopic state is dangerously close to simply defining "the problem" (my
viewpoint) away.

I don't mind defining it away.

How much of what my guage measures is the macroscopic state and how much is
the microscopic state?? The dividing line seems rather arbitrary, unless
you simply answer the question by saying, "that which is different from the
average value, isn't measuring pressure (but rather microstate info); and
that which is the same as the average, is measuring pressure and macrostate
info".

The definition of the macrostate *is* somewhat arbitrary. For a closely
monitored/highly resolved system the macrostate is more restrictive than one
which is only defined in terms of a few macroscopic average parameters. The
closely monitored/highly resolved system has (a very little) less entropy
than the less carefully observed system because the more closely monitored
system has more known about it than the loosely defined system has, and there
is, correspondingly, less ignorance about its possible microstate.

When a sensitive/fast response pressure gauge registers Brownian
fluctuations and the reading of the gauge is used to restrict the range of
possible microstates (such as by observing some of the patterns in the
shuffles in the card deck analogy) then such a restrictive macrostate
definition somewhat changes (slightly) the calculated expected statistical
properties of the system. For a given gauge resolution which measures the
local pressure at the sensing surface over a given timescale a particular
coarse-grained pressure may be operationally defined by the gauge's
spatio/temporal resolution and precision. Such a coarse-grained pressure
corresponds to a local average pressure for the local subregion of the system
interacting with the gauge. If the gauge's response time is much faster than
the equilibration time for the spatial region of the system which is in
measurable contact with the gauge, then the gauge will not necessarily be
reading the subregion's local (average) pressure but will be looking at the
microscopic fluctuations for that subregion. If the gauge response time is
much slower than the equilibration time for the subregion then the gauge may
measure the subregions's local pressure. In this case local effective
ergodicity can be used to relate the partially time-averaged gauge reading to
the statistical (mech) average phase-averaged properties for the subregion of
the system in contact with the gauge as long as the local average pressure is
changing on a sufficiently slow time scale as the subregion interacts with
other subregions whose local average pressures are somewhat different in
local value. (This is the approximation of 'local equilibrium'.)

The problem I have with that is the following: what if I don't know
if the system is in equilibrium and my pressure guage sees fluctuations, are
the fluctuations fluctuations in pressure (because its not in equilibrium);
or are they measuring micro info (because my system is in equilibrium and
then the fluctuations aren't pressure??)???

See the comment above. Typically it's a question that turns on the
spatio-temporal time scales of the fluctuations and the local equilibration
rates at various spatial scale sizes.

This may of course be the difference implicit in taking the information
theory approach versus the other (I'm not sure).

Joel

I think it is more a question of just how detailed one wants to be in one's
definition of the macrostate.

David Bowman
dbowman@gtc.georgetown.ky.us