Consider the fundamental statistical definition of entropy, as stated by boltzmann and gibbs, s=-kb p(n) log[p(n)] for a probability distribution p(n) over a discrete set of micro states n = 1,. the boltzmann constant prefactor is not fundamental. it is only there to ensure the value of s is of order one when the number of particles is of order na, since kb ~1/na. google: "boltzmann", "gibbs", "statistical-entropy", "h-theorem", and "shannon en- tropy". in information theory some people seem to live under the misconception that shannon invented it all in 1945.
a. check that this entropy is equal to zero in case the probability distribution p(n) is equal to zero for all micro states except for one, p (no) = onno (the deterministic, lowest possible uncertainty case).
b. show that in case the distribution is totally flat, p(n) = 1/w (the maximum uncertainty case which maximizes the disorder), the entropy is equal to the formula engraved on boltz- mann's grave. (find it on the
c. what is the entropy of the atmospheric law in problem 19? what is wrong with this question without specifying the spatial resolution (without discretizing space)? what is the entropy of the atmospheric law distribution if we assume a limit in spatial resolution of 1 meter?
d. is the atmosphere in a state of maximum entropy?