**CHAPTER 7**

**DIFFICULT COUNTING**

The tomb of Ludwig Boltzmann in Vienna is engraved with a very short and simple-looking equation, which, ironically, he never wrote down during his lifetime:

*S* = *k* log *W*.

*S* is the universal symbol for entropy, *k* is a fundamental constant of nature known as Boltzmann’s constant, and log *W* is the logarithm^{1} of a number, *W*, relating to the physical system of interest, a number that Boltzmann called the number of “complexions.” (The number *W* can be so devilishly hard to calculate for many physical systems of interest that the greatest mathematical physicists of the twenty-first century, and the most powerful computers as well, are helpless to determine it.) This equation was the lever for setting the quantum revolution ...

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