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Two level system permutations

  1. Nov 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Sounds like a physics problem but I'm sure of the physics, stuck on the maths. At high T a two level system has ##\frac{N}{2}## particles in each level. If entropy is given by ##S = k\ln(\Omega)##, where ##\Omega## is the number of ways of getting ##\frac{N}{2}## particles per level, show the high temperature limit is ##Nk\ln(2)##.

    2. Relevant equations


    3. The attempt at a solution
    To the best of my knowledge, ##Omega = \frac{N!}{(\frac{N}{2})!(\frac{N}{2})!}##. Taking ##ln## of this and using the Stirling approximation:

    ##N\ln(N) - N - [\frac{N}{2}\ln(\frac{N}{2}) - \frac{N}{2}] - [\frac{N}{2}\ln(\frac{N}{2})-\frac{N}{2}]##
    ##= N\ln(N) - N\ln(\frac{N}{2})##
    ## = -Nln(2)##
    I've gone wrong with a minus sign somewhere but I really can't see where! Thanks for any help!
     
  2. jcsd
  3. Nov 16, 2016 #2

    DrClaude

    User Avatar

    Staff: Mentor

    How do you write ##-N \ln(N/2)## as a sum of two logarithms?
     
  4. Nov 16, 2016 #3
    Oh yes, there's the missing minus sign. Thank you! :)
     
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