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Entropy

``If we wish to find in rational mechanics an a priori foundation for the principles of thermodynamics, we must seek mechanical definitions of temperature and entropy.'' -- J.W. Gibbs

The function   $\Omega(n,N)$  is called the MULTIPLICITY FUNCTION for the partially specified system. If  N  and  n  get to be large numbers (which is usually the case when we are talking about things like the numbers of electrons in a crystal),   $\Omega(n,N)$  can get really huge.^15.8 It is so huge, in fact, that it becomes very difficult to cope with, and we do what one usually does with ungainly huge numbers to make them manageable: we take its logarithm. We define the [natural] logarithm of  $\Omega$  to be the ENTROPY  $\sigma$:

$$\sigma \; \equiv \; \ln \Omega$$ (15.3)

Let's say that again: the ENTROPY  $\sigma$  is the natural logarithm of the MULTIPLICITY FUNCTION  $\Omega$  -- i.e. of the number of different ways we can get the partially specified conditions in this case defined by  n.

Is this all there is to the most fearsome, the most arcane, the most incomprehensible quantity of THERMODYNAMICS? Yep. Sorry to disappoint. That's it. Of course, we haven't played around with  $\sigma$  yet to see what it might be good for - this can get very interesting - nor have I told this story in an historically accurate sequence; the concept of ENTROPY preceded this definition in terms of ``statistical mechanics'' by many years, during which all of its properties were elucidated and armies of thermal physicists and engineers built the machines that powered the Industrial Revolution. But understanding THERMODYNAMICS the old-fashioned way is hard! So we are taking the easy route - sort of like riding a helicopter to the top of Mt. Everest.