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Deriving the generalized entropy function

  1. Nov 26, 2011 #1
    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
     
  2. jcsd
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