What is the derivation for S = k log ⁡ N, where k is Boltzmann constant, N number of microenergy states.How is the logarithmic measure justified?

Question

What is the derivation for   S  =  k  log  ⁡  N, where k is Boltzmann constant, N number of microenergy states.How is the logarithmic measure justified?

Xzavier Barr · Accepted Answer

The best way to justify the logarithm is that you want entropy to be an extensive quantity -- that is, if you have two non-interacting systems A and B, you want the entropy of the combined system to be      S          A      B        =      S    A    +      S    B    .If the two systems have       N    A    ,      N    B   states each, then the combined system has       N    A        N    B   states. So to get additivity in the entropy, you need to take the log.You might wonder why it&#039;s so important that the entropy be extensive (i.e., additive). That&#039;s partly just history. Before people had worked out the microscopic basis for entropy, they&#039;d worked out a lot of the theory on macroscopic thermodynamic grounds alone, and the quantity that they&#039;d defined as entropy was additive.Also, the number of states available to a macroscopic system tends to be absurdly, exponentially large, so if you don&#039;t take logarithms it&#039;s very inconvenient: who wants to be constantly dealing with numbers like       10                  10                  20                     ?

What is the derivation for S=klogN, where k is Boltzmann constant, N number of microenergy states.

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