# Deriving the generalized entropy function

1. Nov 26, 2011

### 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 possess 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