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]

Re: [Phys-l] equilibrium, probability, and Boltzmann factors



Hi all-
To the extent that this is a non-equjilibrium situation, it is not discussed in most texts currently in use; see the introductory comments in my paper on the ArXiV 0806.012 (phys-gen) also accessible through Google-scholar under - Uretsky,J ising -

"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley




On Tue, 20 Apr 2010, John Denker wrote:

Hi Folks --

Suppose we have an overall system consisting of two
subsystems, A and B. The overall system is closed
and isolated, so its energy is constant. However,
subsystem A can exchange energy with subsystem B.

Suppose subsystem B is large enough and well behaved
enough to serve as a heat bath. OTOH subsystem A
is much smaller, consisting of just two spin-1/2
particles (four microstates, three energy levels).

We let the system sit for a long time, so that it
reaches its maximum entropy macrostate. This is what
I call thermal equilibrium, although I don't want to
argue about definitions.

Please consider the following hypothetical statements
and mark them as T or N, where
-- T means reliably true, to a good approximation,
for all practical purposes, and
-- N means not reliably true; this includes statements
that are sometimes true and sometimes not.

As for the overall system:
T/N -- All the microstates have the same energy and
therefore they are all equally probable.

As for subsystem A:
T/N -- All the microstates are equally probable.
T/N -- Any two microstates that have the same energy
will have the same probability.
T/N -- Any two microstates that differ in energy will
have probabilities that are related by a Boltzmann
factor; specifically, Pi/Pj = exp((Ej-Ei)/kT)
T/N -- The probability distribution does not even
remotely resemble a Boltzmann distribution.

Ditto for subsystem B:
T/N -- All the microstates are equally probable.
T/N -- Any two microstates that have the same energy
will have the same probability.
T/N -- Any two microstates that differ in energy will
have probabilities that are related by a Boltzmann
factor; specifically, Pi/Pj = exp((Ej-Ei)/kT)
T/N -- The probability distribution does not even
remotely resemble a Boltzmann distribution.

Remark: These issues obviously have great practical
importance and are central to any real understanding
of thermodynamics.

Hint: Be careful. I know some very smart people who
got some of these questions wrong ... and expressed
100% or near-100% confidence in the wrong answers.
I'm not talking about obscure or trivial exceptions;
I'm talking about categorically, significantly,
provably wrong answers.

Hint: The discussion starting on page 2 of Feynman
_Statistical Mechanics_ is wrong several times over.

The discussion at the beginning of chapter 3 in
Kittel & Kroemer _Thermal Physics_ contains at
least one categorically false statement, and in
other places depends on some undocumented assumptions.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l