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Multiplicity of Macrostates, involving dice

  1. Sep 18, 2011 #1
    I didn't really know where the proper place was for this, but this is an intro thermodynamics class and I'm really confused over this math question (it's not strictly physics-related).

    1. The problem statement, all variables and given/known data

    Consider rolling N six-sided dice. Define a microstate as the number showing on any given die, and the macrostate be the sum across all of the dice. Let n be the macrostate. Find a general formula for the multiplicity of any pair (N, n).

    2. Relevant equations

    I don't know if it's relevant or not, but I know that for N oscillators in an Einstein solid and q units of energy, the multiplicity of (N, q) is (q+N-1) choose q.

    3. The attempt at a solution

    I know a few features the formula must have. If n < N or n > 6N, it must resolve to 0. If n = N or n = 6N, then it must resolve to 1. I know it has to be symmetrical about 3.5N. I have a chart for N = 2, so I know all of the values for all pairs (2, n). I have to imagine that it's related to probabilities, so it's presumably some combination of factorials, permutations, and/or combinations.

    I tried to treat it as an Einstein solid with n units of energy. Knowing that each die has to have at least 1 unit of energy, that means that the q in the formula given would actually be n-N. This didn't give me the right answer though since it allows a die to have a value of n that's greater than 6. I just am having a really tough time figuring out how to build a formula to match the constraints that I know it must have, and at this point I'm completely lost.

    Thanks for the help :)
     
  2. jcsd
  3. Sep 18, 2011 #2
  4. Sep 18, 2011 #3
    For large N I think there is a geometrical solution to this, see the attached.
     

    Attached Files:

  5. Sep 19, 2011 #4
    Sorry, I don't see how any of this is supposed to help me. I just don't understand what you're getting at.
     
  6. Sep 20, 2011 #5
    I'm sure your professor would be more than happy to help! ;0)

    -- Prof. Lyman
     
  7. Sep 20, 2011 #6
    This problem is pretty hard and I think post #3 has the right idea. Perhaps it is easier to calculate the "volume" under the bounding surface (In N = 2, "volume" is area and "bounding surface" is a line) and then you can differentiate w.r.t. n). Your problem will be to evaluate N-dimensional volumes. I think you can use some recursion.
     
  8. Sep 22, 2011 #7
    Hahahaha...this is great.
     
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