Deriving the generalized entropy function

[jonphysics]

Hi,
I'm a graduate student in the life sciences seeking to use entropy maximization to describe ecosystem processes. I have a decent understanding of why S= -k Ʃ pi ln(pi) is a generalized form of S= k ln W, but get stuck in the algebra. Maybe I'm going about it the wrong way.

S= -k Ʃ pi ln(pi)

to

S= k ln W (or S=klnΩ)

when all microstates possesses the same probability.
I'm assuming one needs to invoke:

ln W = NlnN - ƩNi ln Ni.

here is what I did:

S = -k Ʃ Ni/N*ln(Ni/N) where N1 = N2 = N3 ... = Nn

proceeding with the assumption that the set of 'Ni' can be replaced with 'n'

S = -k (n/N*ln(n/N) + n/N*ln(n/N)...)
S = -k n/N( Ʃ (ln(n) - ln(N))

given that there will be N terms, I thought it would be safe to assume that:
Ʃ (ln(n) - ln(N) = Ʃln(n) - N*ln(N)

yielding

S = -k/N(nƩln(n) - nN*ln(N))

Getting me very close to the values required to substitute lnW...

My second notion was to try using

Ni/N=e^(-εβ)/Z

 

[Future Bruno]

= piThis gets me close to the same place, however I'm still not able to get the desired result.Any help would be hugely appreciated!You are almost there. The trick is to use the fact that all of the Ni's are equal:S = -k Ʃ Ni/N*ln(Ni/N) where N1 = N2 = N3 ... = NnBy summing over the same value multiple times, you can simplify your expression:S = -k (N/N*ln(N/N) + N/N*ln(N/N)...)Since there is a factor of N in each term, this simplifies to:S = -k N/N*ln(N/N) And thus the final result is:S = -k ln(N/N) which is equivalent to S = k ln W