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How can one prove that the maximum entropy occurs

  1. Dec 8, 2007 #1
    1. The problem statement, all variables and given/known data
    I am calculating entropy using the formula:

    [tex] S=-\sum_i P_i \ln{P_i} [/tex]

    where the sum is over all of the microstates of my system and P_i is the probability to find a particle in microstate i.

    How can one prove that the maximum entropy occurs when P_i is the same for all i?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 8, 2007 #2

    Avodyne

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    Enforce [itex]\sum_i P_i=1[/itex] with a Lagrange multiplier; that is, extremize
    [tex]-\sum_i P_i\ln P_i + k\bigl(\sum_i P_i-1\bigr)[/tex]
    with respect to both [itex]P_i[/itex] and [itex]k[/itex].
     
  4. Dec 8, 2007 #3

    Dick

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    The sum of the P_i=1, that's a constraint. Add a lagrange multiplier to enforce the constraint, like alpha*(sum(P_i)-1). Now take the partial derivatives wrt all variables and set them equal to zero. The partial wrt alpha gives you sum(P_i)-1=0. The partial wrt to P_i gives you an expression for P_i in terms of alpha. alpha is a constant, so all P_i are equal.
     
  5. Dec 8, 2007 #4

    Dick

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    Great minds think alike. But some are faster.
     
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