9780192806727.pdf

where W 0 is the equilibrium value of W 0. Equations 4.22 and 4.24 show that W 0

is Gaussian in &Ea. Denote (as before) the mean square deviation of this distri-

bution by (e^2 ). Then (e^2 ) = kcgT^2 , which is again Eq. 4.13.

As we now know, although it was not at once clear then, in the early part of
the twentieth century, physicists concerned with the foundations of statistical
mechanics were simultaneously faced with two tasks. Up until 1913, the days of
the Bohr atom, all evidence for quantum phenomena came either from blackbody
radiation or from specific heats. In either case, statistical considerations play a key
role. Thus the struggle for a better understanding of the principles of classical
statistical mechanics was accompanied by the slowly growing realization that
quantum effects demand a new mechanics and, therefore, a new statistical
mechanics. The difficulties encountered in separating the two questions are seen
nowhere better than in a comment Einstein made in 1909. Once again complain-
ing about the complexions, he observed, 'Neither Herr Boltzmann nor Herr
Planck has given a definition of W [E24]. Boltzmann, the classical physicist, was
gone when these words were written. Planck, the first quantum physicist, had
ushered in theoretical physics of the twentieth century with a new counting of
complexions which had absolutely no logical foundation whatsoever—but which
gave him the answer he was looking for.* Neither Einstein, deeply respectful and
at the same time critical of both men, nor anyone else in 1909 could have foreseen
how odd it would appear, late in the twentieth century, to see the efforts of Boltz-
mann and Planck lumped together in one phrase.
In summary, Einstein's work on statistical mechanics prior to 1905 is memo-
rable first because of his derivation of the energy fluctuation formula and second
because of his interest in the volume dependence of thermodynamic quantities,

• Planck's counting is discussed in Section 19a.

74 STATISTICAL PHYSICS

where the expressions in brackets refer to equilibrium values. The first-order

terms cancel since A£ 0 = —AEi (energy conservation) and [dS 0 /dE 0 ] = [dSJ

6>£,] (equilibrium). Furthermore, [3*S 0 /dEt] = -l/c 0 T^2 and [d*SJdE\} =

— \/CjT^2 , where c 0 ,c^ are the respective heat capacities at constant volume and

c, » c 0 since F, » V 0. Thus Eq. 4.22 becomes

Next Einstein applied the relation S 0 = k In W 0 to the subsystem and reinter-

preted this equation to mean that W 0 is the probability for the subsystem to have

the entropy S 0 (at a given time). Hence,

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