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Statistical mechanics - Partition function of a system of N particles

  1. Jan 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Imagine a system with N distinguishable particles. Each particle may be in two states of energy: -ε and +ε.

    Find the the partition function of the system

    2. Relevant equations

    2acad120bcee798b08a9cdfca4db8451.png

    3. The attempt at a solution

    I know that I have to find the partition function for a single function, Z, and my final result will be ZN. Now, I'll say that:

    (Where it says ε it's meant to be ε(r) )

    Z = Ʃr exp(-β(ε - ε) ) = Ʃr exp(-βε) * exp(βε) =
    = Ʃr exp(-βε) * Ʃr exp(βε)



    I'm sure this is incorrect. It doesn't make sense in my head.. E(r) is the energy associated with each microstate, therefore saying that E(r) = ε(r) - ε(r) can't make any sense! I know that the result is:

    Z = ( exp(βε) + exp(-βε) )N

    I have no idea how to get there tho. How did it became a sum? How do I get rid of the summatories?

    Any help will be appreciated!
    Thanks.
     
  2. jcsd
  3. Jan 2, 2013 #2
    The partition function is a summation over states. You simply are using the summation wrong. It is not a summation over the energy levels of within the exponent. It is a summation over e(-Es/T).
     
  4. Jan 2, 2013 #3
    Look at any example problem in a thermo book for a 2-state system
     
  5. Jan 3, 2013 #4
    8ikmAm I not summing over the expoent of the energy of each microstate?

    Do you know any good statistical mechanics book you'd advise me reading?

    EDIT:
    Is it a summation over all the states of energy instead of the energies of each microstate? Because then the solution would make sense!
     
    Last edited: Jan 3, 2013
  6. Jan 3, 2013 #5
    No. You are summing over the exponential function raised to the -Es/T.

    A good book is Thermal Physics by Kittel + Kroemer
     
  7. Jan 3, 2013 #6
    Where Es is the energy of the s-th state
     
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