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Multiplicity function of the two state paramagnet

For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N/2.

(a) Use Stirling's approximation to estimate the height of the peak in the
multiplicity function.

(b) Derive a formula for the multiplicity function in the vicinity of the peak.

(c) How wide is the peak in the multiplicity function?

(d) Suppose you flip 1,000,000 coins. Would you be surprised to obtain 501,000
heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and
490,000 tails? Explain.

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Some comments (the solution is in the attached file):

In this problem we keep the sqrt[2 pi N] factor in Stirling's approximation. I'm also doing (a) and (b) at once. I don't ...

Solution Summary

The multiplicity of a two state paramagnet is Omega = (n1 + n2)!/(n1! n2!) where n1 and n2 are the number of up and down spins, respectively. In terms of the total number of spins N = n1 + n2 and the excess number of up spins eta = n1 - N/2, this can be written as Omega = Binomial[N, N/2 + eta]. Using Striling's formula, we derive an expression for Omega valid for large N and small eta/sqrt(N).