ENTROPY AND PROBABILITY 731915: 'His discussion [of the second law] is rather lengthy and subtle. But the
effort of thinking [about it] is richly rewarded by the importance and the beauty
of the subject' [E43]. Of Gibbs he wrote in 1918: '[His] book is ... a masterpiece,
even though it is hard to read and the main points are found between the lines'
[E54]. A year before his death, Einstein paid Gibbs the highest compliment. When
asked who were the greatest men, the most powerful thinkers he had known, he
replied, 'Lorentz,' and added, 'I never met Willard Gibbs; perhaps, had I done so,
I might have placed him beside Lorentz' [Dl].At the end of Section 4a, I mentioned that Einstein preferred to think of prob-
ability in a phenomenological way, without recourse to statistical mechanics. The
final item of this chapter is an explanation of what he meant by that. To begin
with, it needs to be stressed that Boltzmann's principle was as sacred to Einstein
as the law of conservation of energy [E54]. However, his misgivings about the
way others dealt with the probability concept led him to a different way, uniquely
his own, of looking at the relation between S and W. His proposal was not to
reason from the microscopic to the macroscopic but rather to turn this reasoning
around. That is to say, where Boltzmann made an Ansatz about probability in
order to arrive at an expression for the entropy, Einstein suggested the use of
phenomenological information about entropy in order to deduce what the proba-
bility had to be.
In order to illustrate this kind of reasoning, which he used to great advantage,
I shall give one example which, typically, is found in one of his important papers
on quantum physics. It concerns the fluctuation equation 4.13, which had been
derived independently by Gibbs and by Einstein, using in essence the same
method. In 1909, Einstein gave a new derivation, this one all his own [E24]. Con-
sider a large system with volume V in equilibrium at temperature T. Divide V
into a small subvolume F 0 and a remaining volume F,, where V = V 0 + F,, F 0
<K F,. The fixed total energy is likewise divided, E = E 0 + Et. Assume* that
the entropy is also additive:
"This assumption was briefly challenged at a later time; see Section 2la.Suppose that E 0 , Et deviate by amounts &E 0 , A£, from their respective equilib-
rium values. Then