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

New Trends in Physics Teaching IV
change to a system, which can be imagined happening, not a formal play with permutations
(though above it was cast in a generalized way, it need not be). Incidentally, it avoids the use of
Stirling’s approximation as well.
Distributions: most probable and average
As we saw, Gurney uses Wmax and so the most probable distribution; not S2 and the average
distribution. This creates the problem that sooner or later he and others who take that approach
have to argue that A In Wmax A In S2, since they really need AS = k In S2.
More seriously, as Gurney himself brings out very well, it is only changes in 52 in an interacting
system that have any interpretation. That is, if parts X and Y of a system are independent, then
‘X = ’ total
but the same is not true for W,,,. Thus even to introduce temperature on Gurney’s approach
(through thermal equilibrium between two systems), S2 has to be used.
Variations on Gurney
Powles extends the informal introductory use of Gurney’s arguments by numerical examples
(Powles, 1968). He calculates by enumerating states the most probable and the average distributions,
going so far as to calculate the specific heat capacity of a five-oscillator Einstein solid. Such
examples seem likely to be of considerable value in developing an intuitive idea of the subject,
and must be welcome in a topic nomially ridden with algebra and short on reality.
Sherwin offers a much more radically worked out version of Gurney’s arguments, also (unlike
Powles) exploiting the unit-change method. Like Powles, he uses extensive numerical examples,
enumerating states. In these examples, temperature is given a qualitative statistical sense, as a
measure of the probable direction of spontaneous energy flow, since the examples include two
systems coming to thermal equilibrium.
He also uses W, rather than CL, and this means that he has to do a difficult sum over an
infinite number of exponential terms to obtain the specific heat capacity. This calculation is
much easier with S2 (cf Bent 1965, Nuffield Advanced Physics, 1972). However, it is in Sherwin
that one major advantage of the Einstein solid is clear. Given that the arguments lead to an
exponentially graded distribution, temperature can be introduced as related to the gradient:
the hotter the solid, the shallower the grading, so T appears as a divisor in exp(-E/kT).
BENT’S APPROACH
Bent deserves a discussion of his own, because, although like Gurney he uses the Einstein solid
and unit-change arguments, he uses them to operate on S2 rather than Wmax, and because he
offers a total pattern of teaching which has flair, is distinctive, and has some rare merits.
The boxed parts of figure 8 show Bent’s statistical arguments, to be compared with figure 7
(Gurney). He has a unique, if not totally convincing, argument by induction from small numbers
to the expression
S2 = (N-l+q)! /(N-l)! (q)!
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