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Re: [Phys-l] Temperture profile in a graviational field



I agree with about 98% of what Christopher M. Gould wrote on
01/17/2012 03:46 PM ... everything except the bottom-line
conclusion.

The discussion of "mass transport equilibrium" is worth pursuing,
and may hold the key to answering the original question.

In particular, this makes it clear that in *my* earlier note,
even though I stand by the conclusion, the other 98% of my
note was alarmingly slipshod. Specifically, at one point I
defined inverse temperature in terms of dS/dE ... but really
I should have written ∂S/∂E ... and then we must invoke the
rule that requires writing ∂S/∂E in the direction of constant
"something", and the direction should always be spelled out.
There is a tradition in thermodynamics of being sloppy about
the directions, and it leads to endless trouble. Don't do it.
Always spell out the directions of your partial derivatives.
The time it takes to spell out the directions is nothing
compared to the time it takes to repair the damage that
sloppiness causes.


Sooooo ... If we do things properly: For parcels that can
exchange energy and nothing else, we write
β = ∂S/∂E at constant V

Meanwhile, for parcels that can exchange particles as well
as energy, we write:
β = ∂S/∂E at constant V and constant N

This is important, because the original question described
a situation where N is changing.

One way of describing the physics is that fluctuations in
N will be correlated with fluctuations in T, in the scenario
suggested by the original question. To find the actual
temperature, we can keep the scenario and correct for the
N-dependence mathematically ... or we can consider a slightly
modified scenario that keeps N constant, to save us from
doing the math.