# Multiplicity of Macrostates, involving dice

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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Spinnor
Gold Member
Spinnor
Gold Member
For large N I think there is a geometrical solution to this, see the attached.

#### Attachments

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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.

I'm sure your professor would be more than happy to help! ;0)

-- Prof. Lyman
Hahahaha...this is great.