Multiplicity of Macrostates, involving dice

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The discussion revolves around finding a formula for the multiplicity of macrostates when rolling N six-sided dice, where a microstate is defined by the outcome of each die and the macrostate is the total sum. Key points include the requirement that the multiplicity must be zero for sums less than N or greater than 6N, and it should equal one for sums equal to N or 6N. The conversation suggests that the problem may relate to probabilities and involves combinations or permutations, with hints towards using geometric solutions or recursion for larger N. Participants express confusion over how to construct the formula while adhering to these constraints. Overall, the problem highlights the complexities of calculating multiplicities in statistical mechanics.
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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).

Homework Statement



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).

Homework 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.

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 :)
 
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For large N I think there is a geometrical solution to this, see the attached.
 

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

-- Prof. Lyman
 
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.
 
boltz_man said:
I'm sure your professor would be more than happy to help! ;0)

-- Prof. Lyman

Hahahaha...this is great.
 
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