New trends in physics teaching, v.4; The ... - unesdoc - Unesco

New Trends in Physics Teaching IV
do arguments about changes in numbers of microstates appear, nor is dS = kd (In a) to be found.
Is he then ‘doing statistical mechanics’? Yes, in the sense that the Boltzmann factor contains the
statistics. No, in the sense that it is not clear how it does so.
Feynman can finesse the statistics, because of the form common to all the applications he
discusses, indeed to any. Briefly, any process that borrows energy E from a heat bath involves
an entropy decrease of the heat bath of AS = - E/T (leaving aside qualifications about constant
volume etc.). Writing this as A In 52 = - E/kT, and A In 52 as In ( after/a before) we always get
The ratio of the values of 52 after and 52 before borrowing energy E gives the probability that it
can be borrowed, and so the numbers of particles able to borrow that much energy. Thus the
Boltzmann factor works because it is a ratio of numbers of microstates.
The finesse shows up in what looks like a trivial point in Feynman’s argument. The gas must
be isothermal, so he introduces a ‘copper bar’ to keep it so (figure 2). Now of course the Boltzmann
factor does not just apply to gases under gravity (the examples Feynman uses are to show this).
So the argument is not really involved with the system being a gas, even though it appears to be
all about that. The ‘copper bar’ is what does the trick. The energy rngh for a molecule to get up a
height h cannot come from the gas if the gas must not become cooler in any part. It must come
from the ‘copper bar’ heat sink; indeed, this is just what a canonical ensemble argument relies
on. There are fewer molecules at great heights because the sink is unlikely to yield up that much
energy by a chance fluctuation. There are fewer molecules at great heights because the ones
below must hold them up, too, but the two ‘becauses’ are very different. The first is an essentially
statistical mechanical argument; the second is particular and ad hoc. Feynman uses the second
and not the first, and so is unable to show why the result he gets is general, not special.
-
Mechanism
for equalizing d
temperatures.
h
Feynman’s ‘copper bar’ to maintain gas
isothermal state
Figure 2. Feynman’s ‘copper bar’ to maintain gas in isothermal state.
Feynman and heat
In speaking of the second law, Feynman says things like, ‘. . . it ought to be possible to lift
weights. . . and thus to do work with heat’; ‘If the whole world were at the same temperature,
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