Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

max entropy - more discussion



David wrote
_________________________________________________________

The definition of the pressure requires that it *is* an averaged quantity.
The microscopic impulses do not define a time dependent pressure function.
Actually, for a hard sphere system the instantaneous value is not well
defined
mathematically since it represents a very discontinuous function whose value
is zero for all times except for a set of times of measure zero, and at
those
times the value is infinite.

The pressure of any (fluid) system is the negative of the *statistical
average* of the partial derivative of each microstate's energy E_r with
respect to the system's volume over the microstate distribution , i.e.
p = - <dE_r/dV> = - SUM_r{p_r*dE_r/dV}. Being an average, the pressure does
not depend on time in equilibrium since the {p_r} distribution is time
independent in this case. If your pressure gauge is so sensitive and has
such
a fast response time that it is actually registering fluctuations, then your
gauge is not registering the pressure; it is registering part of the
microscopics (which is thus not a purely the macroscopic quantity that the
pressure is supposed to be).
_____________

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. 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.

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 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??)???

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

Joel