Chapter 6.5, Problem 43P

Some advanced textbooks define entropy by the formula $S=-k\underset{s}{\sum }P\left(s\right)\ln P\left(s\right)$, where the sum runs over all microstates accessible to the system and P(s) is the probability of the system being in microstate s.

1. For an isolated system, $P\left(s\right)=1/\Omega$ for all accessible states s. Show that in this case the preceding formula reduces to our familiar definition of entropy.
2. For a system in thermal equilibrium with a reservoir at temperature T, $P\left(s\right)=e^{-E\left(s\right)/kT}/Z$. Show that in this case as well, the preceding formula agrees with what we already know about entropy.